My previous answer focused on what might be called the prehistory of "top of the morning"—the period between about 1650 and about 1700 when the expression was used descriptively rather than as a salutation.
This answer instead looks at published instances where "top of the morning" is used as a greeting or pleasantry. In particular, I want to look at instances where the expression is put in the mouths of Irish characters and instances where it is attributed to non-Irish characters.
The starting point for such usage remains George Walker, Theodore Cyphon: Or, The Benevolent Jew (1796)—meaning that a century of silence separates the last descriptive instance of "top of the morning" (John Partridge in 1693) from the first welcoming instance (George Walker in 1796).
'Top of the morning' as a salutation in non-Irish contexts
After the Theodore Cyphon instance of "the top of the morning to you" from 1796, the next two Google Books matches for the phrase used as a salutation are from the works of Walter Scott. These appear in books published in 1815 and 1818, as follows. From Guy Mannering (1815), spoken by a ship's captain named Dirk Hatternick:
The fellow's cloudy visage cleared up. "The top of the morning to you, sir; I find you are a visitor of my friend Mr Bertram—I beg pardon, but I took you for another sort of person."
Hatternick evidently understands Gypsy cant, but his name is vaguely Dutch and Scott describes him as having "somewhat of a foreign accent, though speaking perfectly good English." It is also noteworthy that he is mariner, as we shall see.
From Tales of My Landlord [The Heart of Mid-Lothian] (1818):
As our traveller set out early in the ensuing morning to prosecute her journey, and was in the act of leaving the inn-yard, Dick Ostler, who either had risen early or neglected to go to bed, either circumstance being equally incident to his calling, hollowed out after her, "The top of the morning to you, Moggie. Have a care o' Gunnerby Hill, young one. Robin Hood's dead and gwone, but there be takers yet in the vale of the bever."
As noted in TaliesinMerlin's answer, Dick Ostler is evidently a Yorkshireman—and a landlubber.
From "The Legend of the Stone House," serialized in The Ladies' Literary Cabinet (March 2, 1822):
At length exclaiming, "Shipmate, ho! the top of the morning to you," no answer was returned to this salutation, when he [Hobbs] continued, "where from my friend! what are you doing here?"
Hobbs is a sailor who has nothing Irish about him.
From Lee Gibbons, The Cavalier: A Romance, volume 2 (1822):
The dissimilarity between the manners of the Borderer, and the old courtier-like appearance of his dress, was so evident as to strike all beholders with admiration. On entering the room, he advanced to the ladies, and in his usual hasty manner cried out, "Ah! ah! Mistress Bradshaw! yer servant Madame! Miss Esther! guid mornin', the top of the mornin till ye.—Ye look like the queen o' the fairies, washit i' the dew of heaven."
The dialect portrayed here—as in much of Walter Scott's work—is evidently Scottish Border English.
From an anonymous author, Tales of an American Landlord; Containing Sketches of Life South of the Potomac, volume 1 (1824), speaking in the character of Dr. Clapperton, an American:
"The top of the morning to you, my Lord," said he; ' I saw you an hour gone by brushing away the dew. And so you have already encountered the keen wits of the mad poet of Hopewell? I have perceived you and him yard-arm and yard-arm for the last two glasses, and considered it high time to run down to your assistance. ..."
Dr. Clapperton is presented as a native-born American medical man.
From John Howell, "The Man-of-War's-Man," in Blackwood's Edinburgh Magazine (April 1826):
While the Court was assembling, I took a stroll round the decks, and had not gone far when I was accosted by Joe Green, one of the delegates of the ship, with a familiar slap on the shoulder, coupled with the usual salutation of,—'Aha, matey!—the top of the morning to you, my boy!—Hast any news?'
Here the entire dialogue is between sailors.
From "Nim South in Leicestershire" The Sporting Magazine (November 1830):
The top of the morning to you, Mr. Editor: here I am in the heart of Leicestershire. All the saddlers, livery-stable-keepers, innkeepers, lodging-house-keepers, and other keepers, are expecting to make a fortune by the emigration of that great fox-hound keeper Lord Southampton, from Quorndon Hall to this ancient and independent borough: ...
Nim South is an English sporting enthusiast, evidently with a particular interest in the hunt.
From "General Lee," a vignette set in Watertown, Massachusetts, in The Ladies' Garland (published in Harpers-Ferry, Virginia, October 15, 1835):
Gen. Lee—The top of the morning to you, good woman. Can you give a soldier a draught of milk?
The General Lee here is presumably Henry "Light-Horse Harry" Lee of Virginia, a hero of the American Revolution and the father of Robert E. Lee.
From Michael Scott, "Scenes in Jamaica," in Tom Cringle's Log, serialized in Blackwood's Edinburgh Magazine (October 1832):
"Good morning, Mr. Whiffle—the top of the morning to you, sir."
This line is spoken by the book's title character, Tom Cringle, a young Englishman in the British Navy.
From William Ainsworth, Rookwood: A Romance (1834):
"The top of the morning to you, gentlemen," said Turpin, (for so we must in future designate our friend, Jack Palmer) as he rode up, at an easy canter. "Did you not hear my halloo? I caught a glimpse of you on the hill yonder-I knew you both, two miles off; and so, having a word or two to say to you, Luke Bradley, before I leave this part of the country, I put Bess to it, and she soon brought me within hail. ..."
Turpin is Dick Turpin, the famous English highwayman.
From W.J. Neale, Will Watch. From the Auto-Biography of a British Officer, volume 2 (1834):
"Well, my old lady!" said he [Kerslake, a lieutenant in the British Navy], "the top of the morning to you!"—the "old lady" curtseyed.—"This your daughter, old soul?"—trying to take the hand of the latter, as she retired in much confusion.
Again we have a British Navy connection.
From Frederick Thomas, Clinton Bradshaw; or, The Adventures of a Lawyer, volume 1 (1835):
The sun was up some hours, before Bradshaw left his room; when he did, it was noiselessly, so as not to disturb Selman. ...
"The top of the morning to you, Bradshaw," said Willoughby.
"The top is toppling down towards the meridian," said Cavendish. "But where’s Selman?"
Bradshaw is the titular lawyer of this novel.
These examples stretch from 1796 to 1835 (at which point I cut off my search for examples). A disproportionate number involve people who have worked aboard ships, but there is also significant representation of the Scottish-English Border and the United States. None of the characters in these examples show any sign of intentionally or accidentally slipping into Irish speech patterns in any other respect, which suggests that readers in the period from 1800 through 1835 did not assume that saying "the top of the morning to you" automatically marked the speaker as being Irish.
'Top of the morning' as a salutation in Irish or pseudo-Irish contexts
Recorded instances in which writers attribute the phrase "top of the morning" to Irish speakers begin fairly early (by 1809), but the wave of such attributed usage doesn’t become a tsunami until the period 1829–1835. In gathering this group of examples, I have tried to keep track of whether the author was Irish and whether the piece was published in Ireland.
The earliest match for "top of the morning" as a salutation between Irish characters dates to 1809, in "Irish Courtship," a comic song included in The Hibernian Cabinet; A Selection of All the Most Popular Irish Songs, Than Have Ben Lately Written (published in London, 1817). Here is how the opening of the song is presented in The Hibernian Cabinet:
A favorite comic Duet sung by Mr. Rose and Mrs. Pearce in the comic Pantomime of Fairy Land, or the Wandering Magician, at Astely's Amphitheatre.—1809.
Mr. Rose. Och, the top of the morning to Katty my jewel, / Fait, your eyes, joy, as white as the milk in your pail, / Have bodder'd my heart, and so tipt me my gruel, / That by Paddy’s own self, I don't know what I ail: / ...
The same song also appears in The Universal Songster; Or, Museum of Mirth (1825), with authorship attributed to the English songwriter C.F. Barrett, and again in The Shamrock; A Collection of Irish Songs (published in Glasgow in 1830), a volume notable for the editor's explicit acknowledgment that Irish comic songs tended to be "written by persons who for the most part imagined, that to dress a flat contradiction in rhyme was to make a comic Irish song"; that is, they often involved caricatures of Irish speech by non-Irish writers. Still it is noteworthy that this popular song antedates all but the Theodore Cyphon instance of "top of the morning to you" and appears in the context of purported Irish idiom.
Another early instance appears in William Moncrieff, Wanted a Wife; Or, A Checque on My Banker (1819), in dialogue spoken by a character named Barney M'Shift, who claims to be "descended from the O'Mac's of Trotterbogslough":
M'Shift. Och! here he is: upon my word he seems to have been a mighty high fellow in his time, though he looks a little low in the world at present.—The top of the morning to your honour.
Moncrieff was English.
From Anonymous, "The O'Mullingonian System; Address to the Inhabitants of Liverpool," in The Kaleidoscope (August 17, 1824):
The top of the morning to yees, my darlings! My name's Mr. Terence O'Mulligan, Esquire, and may I never be after seeing that jewel of a place, sweet Tipperary, never no more, if I arent after feeling the highest felicity in addressing your honours, all three both together; for ye look as rosy, be my hod! as our one cow Paddy on a May morning.
This is evidently a specimen of Irish-mocking humor—as some people reckoned humor in 1824.
From "Darby and the Tug-Pin," in [Boston] Masonic Mirror, and Mechanic's Intelligencer (September 23, 1826):
As Darby Croney was perambulating Water-street, one misty morning, in search of work and a dram, he saw at a distance his friend, Michael Fowler, in seeming difficulty with Jenny, his high-blodded plodder—high in bone, I would rather say, than in flesh or in blood.—"The top of the morning to you, Darby," said Michael, "sure, ar'nt I right glad to see you when I'm in trouble now?" "Arrah, the top of the morning to you, Michael : Pray what the divil ails yourself and Jeanie : Are you stuck in a bog?"
The above is a U.S. presentation of Irish stereotypes, and not a friendly one.
From Daniel O'Rourke, "Latest Intelligence from the Moon," in The Inspector, Literary Magazine and Review (1827):
"The top of the morning to your aygleship," was the first salutation of Dan [O'Rourke], who, in his polite waggery, forgot it was barely midnight.
"A fine evening, Mr. O'Rourke," replied the eagle, in very excellent Irish; "I hope Mrs. O'R. and all the little ones are well."
"Fait, an' is it yourself that would spake, and in as proper as English as ever was spoken in Ireland?"
I couldn't find any information on the literary personage Daniel O'Rourke, but the Irish writer Thomas Crofton Croker wrote a book titled Daniel O'Rourke in 1829, and the name seems to be associated with Irish tall tales and fables in which O’Rourke figures as a character.
From the O'Hara Family, "The Rival Dreamers," in The Gem: A Literary Annual (1829):
"No matter, Noreen, only God be wid you, and wid, her, and wid the weenocks; and tell her what I bid you. And more betoken, tell her that poor Shamus [Dempsy] quits her in her trouble, with more love from the heart out, than he had for her the first day they come together ; and 'ill come back to her, at any rate, sooner or later, richer or poorer, or as bare as he went—and maybe not so bare either—only God knows. And so the top o' the morning to you, Noreen, and don't let her want the mouthful of praties while I'm on my travels; for this," added Shamus, as he bounded off, to the consternation of old Noreen,—"this is the very morning, and the very minute that, if I mind the dhrame at all at all, I ought to mind it;—ay, without ever turning back to get a look from her, that 'ud kill the heart in my body entirely."
As noted in a comment beneath my other answer, the O’Hara Family is actually two brothers—John and Michael Banim—of Kilkenny, Ireland. The Banims had ambitions to produce serious Irish literature and would have been in a position to know whether Irish country folk in the 1820s and 1830s commonly used the expression "the top of the morning to you."
From Thomas Crofton Croker, A Sunday in Killarney," in Legends of the Lakes; Or, Sayings and Doings at Killarney, volume 2 (1829):
Amid this state of affairs [a serious drought in Kerry], as Mr. Lynch and myself were enjoying the luxurious shade of the oak wood a decent elderly man was accosted by that gentleman [Mr. Lynch], with "Good morrow, Daniel."
"Oh, the top of the morning to your honour," said he.
According to Encyclopædia Britannica, Croker was an "Irish antiquary whose collections of songs and legends formed a storehouse for writers of the Irish literary revival"; he was born in Cork but moved to England in 1818. This instance is striking because it is presented as a reminiscence about a particular day, and the old man's greeting is mentioned only because he goes on to tell Lynch and Croker a folktale about how God made three ears of corn grow out of a griddle-cake.
From Thomas Grattan ("an Irish miscellaneous writer," according to his Wikipedia entry), "The Love-Draught: A Tale of the Barrow-side," in The Athenaeum and Literary Chronicle (November 18, 1829):
"The top o' the mornin' to you, Biddy [Keenahan]!" said Lanty, with a sort of half-look of mingled kindness and timidity.
"God save ye kindly, both!" was Biddy's almost inaudible reply ; for the faintness of anxiety, the mixture of hope and fear, almost overcame her.
This instance, too, has some credibility as representing actual contemporaneous Irish speech.
From "Irish Priest's Sermon," in The London and Paris Observer (November 22, 1829):
But the Presbytarian got the cleanest fall of all. Where do you think he fell? I suppose, now, you think he fell on his head, or his arm. No such thing—he fell upon the ground. And what do you think he did when he got there? Nivir a single thing to swear by, except lie like a drunken beast on the earth. (The top of the morning to you, Molly Doyle—I hope your early rising will do you no harm.)"
This is a ludicrous presentation of Irish speech by a non-Irish writer.
From the O'Hara Family (again), "The Irish Lord Lieutenant and His Double" in The Englishman's Magazine (August 1831):
"...well if it’s a thing that Captain Lighton axes the Lord Lieutenant to ax ye to cut his harvest, it would only be a good turn, afther all not to refuse; and it may sarve yourselves; and may be it might sarve me, too, in an endayvour I'm goin' to make to get lave to come home from my rambles, and take to arning an honest mouthful again; and so, there's what I'm come a begging to ye for; and now, the good night to ye, boys, or the top o' the morning; for that matter the day's breakin' already. God speed ye."
From "Reminiscences of a Juris-Consult," in The Lady's Book (January 1832):
Some days afterward a rough Patlander opened my office door, and with one of those ducking bows, peculiar I think to his countrymen, accosted me with "The top of the morning to ye'r honour." I raised my head at the odd sound, and returned the salute, but after a more civilized fashion.
The object of this presentation isn’t to make fun of the "Patlander’s" speech patterns, and the vignette is presented as a reminiscence, but the author is not himself Irish and so brings an outsider’s ear to the representation of the visitor’s speech.
From "Paddy the Piper" in Samuel Lover, Legends and Stories of Ireland (published in Dublin in 1832), describing events that occurred at night but with the title character deceived by the moonlight into thinking that it was morning:
So with that he looks up—and what do you think it was, Lord be marciful unto uz, but a corpse hangin' out of a branch of a three.
'Oh, the top o' the mornin' to you, Sir,' says Paddy, 'and is that the way with you, my poor fellow? throth you tuk a start out o' me,' says poor Paddy.
According to the Wikipedia page on him, Samuel Lover was "an Irish songwriter, composer, novelist, and painter of portraits." He was also a grandfather of the composer Victor Herbert.
From "Chit Chat," in The Metropolitan (June 1833):
Captain O'Sullivan. The top of the morning to you, Mr. Editor. If you are not over busy we'll just have a little talk. What do you think of the Whigs now?
This instance has no particular credibility as evidence of how Irish people spoke.
From "All Saints' Eve; or, A Recollection of the Irish Peasantry" by "An American in Ireland," in The Knickerbocker (September 1833):
Well, by gor, thinks I to mysel, not makin it known to nobody, I'll go to father Patt himsel, an if he has a dhrop of blood in his vanes, he'll do me a sarvice—for you see, what made me so bould intirely upon the Sogatrth, was kase I kilt myself savin him, when he was drownded in the say, afther tumblin out o' Mick Mulroony's boat in a squall. So afther biddin the top o' the mornin to his raverince, I ups and I tals him how Kate was the jewel o' my heart, an how she loved me bether nor all the world, an how her father was causin her for to marry, an how the thoughts o' it was killin her intirely ; an sure enough I persuaded him downright how I'd put an ind to my life if I wasn't the man to get her.
This appears to be Washington Irving’s attempt to replicate Irish speech.
From Frederick Marryat, Peter Simple, serialized in Waldie's Select Circulating Library (February 4 [?], 1834):
'Good morning to ye, Mrs. O'Rourke," says I.
'An' the top of the morning to you, Father M'Grath,' says she, with a smile; 'what brings you here? Is it a journey that you're taking to buy the true wood of the cross? or is it a purty girl that you wish to conſess, Father M'Grath? or is it only that you're come for a drop of poteen, and a little bit of chat with Mrs. O'Rourke?'
Marryat was an Englishman, a captain in the Royal Navy, and a novelist. Whatever his gifts at reproducing speech may have been, he was looking at Irish English from the outside.
From Mrs. S.C. Hall, "Jack the Shrimp," from Sketches of the Irish Character, reprinted in Waldie's Select Circulating Library (July 29, 1834):
I saw him one morning, as usual, poking after shrimps ; and was struck by the anxiety and energy of his movements ; notwithstanding his seeming employment he was intently watching every sail that appeared on the blue waters : when he saw me he rapidly approached.
"The top of the morning to ye, young lady, and may every sunrise increase ye'r happiness!"
Hall was an Irish writer, to judge from an introductory note that refers to "the enthusiasm with which the writer dwells upon the fine qualities of her countrymen."
From William Maxwell, "Rambling Recollections.—No. 1," in The Dublin University Magazine (April 1834):
"You are right," said MacDermott, "but it is unfair to let the honest men without, waste time in useless expectation." He opened the window. "Gentlemen of the 'pavé', the top of the morning to ye, as we say in Tipperary. Toddle off if ye plase. I’m going to practice at the post beside ye, and as the light's but indifferent, why, monamondiaoul! I might, by mistake, shoot into the gateway."
Maxwell was an Irish author from County Down, who, after fighting in the Peninsular Campaigns and Waterloo, eventually "took holy orders, obtaining the living of Ballagh, in Connemara".
From a review of Thomas Medwin, The Angler in Wales (1834), in The Monthly Review (October 1834):
"My friend, Sir Ulysses O'Shaunessey, was walking with his lady in the environs of the lake of Killarney, where he was going out salmon-fishing, and met an old beldam, named Mrs. Malwadding.—'The top of the morning to you.'—'The same to you, Mrs. Malwadding.'—'I dramed a drame, your honour.'—'What was your dream about, Mrs. Malwadding?'—'Och I dramed that your honour's honour would give me a pound of tay, and your ladyship's ladyship a shiner.'—'Well, but dreams always are verified by by their contraries.'—'Och, then, it is your honour's honour that's to give me the shiner, and your ladyship's ladyship that's to give me the tay.' Need I tell you that Mrs. Malwadding's wit ensured her both the tea and the money?"—p. 117.
From James Sheridan Knowles, "The Blacksmith of Clonmel," in the New-York Mirror (April 4, 1835):
"The top of the morning to you, William M'Mullin!" cried he [Phil Brennan], as he smilingly stopped and turned round. "The top of the morning to your friends, who have so kindly come along with you to receive me at my mother's house upon my wedding day. Good luck to you all," he added, "and good luck to the honest boys of Clonmel."
According to Wikipedia, Knowles was an Irish dramatist and actor, who left Ireland for Glasgow in 1817 at the age of 58.
From William Maxwell (again), "My Life," in The Dublin University Magazine (August 1835):
At that moment of general distress, a voice from the street exclaimed, 'The top of the morning to you, mother Macnamara! Will you give us “Jolly mortals" again if you please.'
'It's them thieves of the world from the barrack!' exclaimed the butler. 'Open the door and let us out, or, by the etarnal frost, I'll sware my life agin yees in the mornin'!'
From Frederick Marryat (again), Japhet in Search of a Father (1835), in dialogue involving an Irish laborer in an apothecary shop in London:
"By the powers, one would think that you had made a mistake, and handed me the rich man's plaister instead of the poor one. It's less whiskey I'll have to drink any how ; but here's the money, and the top of the morning to ye, seeing as how its just coming on night."
"And sure enough there was a Timothy in the other shop, for I heard the boy call the other by he name ; however, it's no matter, if it took off the skin, it also took off the thumbago, so the top of the morning to you, Mr. Pottykary."
And from the O'Hara Family (again), The Mayor of Wind-gap (1835), a novel set in Ireland:
"The top o' the mornin' to your worship" began Roger [Divey], snatching off his three-cocked hat, scraping his hob-nailed brogues along the pavement, and accompanying this movement by a corresponding bend in his lathy figure.
"A good morning to you, kindly, Roger," answered his worship, his gurgling voice making way with difficulty through his short round throat ; "and what thrials comes on to-day, Roger?"
In my earlier answer, I noted various 17th-century instances of "top of the morning" as a descriptive term. This was followed by almost a century with no matches at all (in Google Books or Hathi Trust databases) for the phrase, and then by an instance in 1796 from an English author of "top of the morning to you" as a salutation.
In the 1800s, the salutation form of the expression proliferated. Some of the earliest (1815–1826) instances do not involve Irish characters or authors. Of these, a disproportionate number involve speech by sailors and navy officers. Others are from the Scottish-English Border region or from the United States. Instances of non-Irish usage continue through 1835 (my research cutoff year); but starting in 1829, the instances involving a specifically Irish context become much more numerous than the other instances taken together.
A number of the instances in which the expression is attributed to Irish speakers come from obviously non-Irish (and in some cases anti-Irish) sources, but others come from native Irish writers and/or appear in Irish publications. The size and the seriousness of this latter group of sources persuade me that "the top of the morning to [someone]" was widely used in Ireland during the 1820s and 1830s.
Still unexplained is how the salutation "the top of the morning to [someone]" arose in the first place. Although Irish-attributed instances go back to 1809, non-Irish-attributed ones go back even farther, to 1796. And then there is the mystery of the discontinuity between the 17th-century descriptive phrase "the top of the morning" and the 19th-century greeting.
Ultimately, it is difficult to identify where and why the salutation "the top of the morning to you" originated, but it seems reasonably safe to conclude that it was in common use in Ireland—and was widely understood in England to be a standard expression in Irish speech—by the 1830s.