The origin of a spoken phrase is seldom simple. That is the case with 'high as an elephant's eye'. Even supposing the origin can be definitively sourced to one person, and that one person offers an origin story, the story is still a story. The truth about the phrase may be more complicated.
I suspect the use of 'high as an elephant's eye' in Oklahoma was sponsored by a common pun in theatrical circles. The pun obliquely refers to a well-known anecdote. Certainly Hammerstein II, as the grandson of the theater impressario Oscar Hammerstein I and the son of theater manager Willie Hammerstein, would have been aware of some version of the story, and would have heard the pun at least one if not many times.
A longer pun, "the corn is as high as an elephant's eye", was probably extemporized by Hammerstein II while composing. The long version plays on at least three meanings of 'corn', in addition to the allusive range of meanings conveyed by the shorter pun, 'high as an elephant's eye'.
One meaning of 'corn', of course, was the common field crop.
Another common meaning of 'corn' at the time of Oklahoma's composition was the colloquial sense of "something corny", where 'corny' signifies
Of such a type as appeals to country-folk; rustic or unsophisticated; tiresomely or ridiculously old-fashioned or sentimental; hackneyed, trite; inferior.
OED attests 'corn' in the 'corny' sense with quotes from 1936, 1937, and 1946.
The third meaning of 'corn' that may have been bubbling toward the surface of Hammerstein's compository stew was the elliptical
The elliptical use of 'corn' to mean "corn-whiskey" is attested in OED by quotes from 1820-1936.
"High", of course, may refer to being drunk. OED attests the sense with quotes from 1846 and 1943 (among other dates).
The "elephant's eye" may be an obscure ('obscure', that is, to folks not steeped in stage lore) reference to a mid-19th century anecdote about a specific theatrical production. The anecdote itself refers to an earlier expression, still common in another form (wherein the elephants are pink): 'seeing the elephant'.
Putting the stage folklore aside for the moment, one meaning of the phrase 'seeing the elephant' is recounted in this humorous report from the 1846 Pickings from the portfolio of the reporter of the New Orleans Picayune, by Dennis Corcoran (bold emphasis mine):
SEEING THE ELEPHANT.
A hard looking case was Jim Griswell as he stood up yesterday before the Recorder, to answer to the charge of being found gloriously corned the previous night. ....
"Griswell," said the Recorder, "you have been found drunk."
"Squire," said Jim — and his eye showed a desire to assume the melting mood, "...I went on a regular wake snakes sort of a spree, and I went here and there, turnin' twistin' and doublin' about, until I didn't know where or who I was...but, squire, I'll say no more, I've seen the elephant...."
The Recorder let Jim Griswell off on his parole, as he confessed he had seen the elephant!
So, 'seeing the elephant' (and variants) was at the time a euphemism for being blind drunk, on a "wake snakes" spree, "gloriously corned". 'Corned', by the way, is a slang term for "intoxicated", as attested by four quotes from the 19th century in OED.
One version of the anecdote from theatrical folklore, here reproduced from the Illinois Journal (Springfield, Ill.) 23 December 1847, p. 1 (bold emphasis mine), builds on the "blind drunk" meaning of 'seeing the elephant', rather than the diversionary offshoot mentioned:
SEEING THE ELEPHANT.
BY FRANK WEBBER.
This term though very common, is perhaps understood by few — that is the origin thereof. It is now applied to young men who pack up their duds, shoulder a musket, and go to revel in the halls of the Montezuma's, or plod on their weary way to Santa Fe. And also those who have got their eye teeth cut for the first time, by coming in contact with the knowing ones of some of the 'Fancy' — Though it is particularly adapted to such, yet the original use thereof was far more racy than its application in modern days. Could I but paint scenes peculiarly rich with a pen as well as Hogarth could with a pencil, even you, kind reader, would acknowledge that seeing the elephant was a sovereign cure for the blues. Wemyss, in his "twenty-six years of an actor's and manager's life," forgot to portray it, and so what my poor pen can do must suffice.
Some years since at one of the Philadelphia Theatres, a pageant was in rehearsal in which it was necessary to have an elephant. No elephant was to be had. ... Days past in the hopeless task of endeavoring to secure one, but at last Yankee ingenuity triumphed, as indeed it always does; and an elephant was duly made to order, of wood, skins, paint and varnish...two of the 'supes' were duly installed as legs. Ned C—, one of the true and genuine 'b'hoys' held the responsible station of 'fore-legs,' and for several nights he played that heavy part to the entire satisfaction of the manager and the delight of the audience. The part, however, was a very tedious one, as the elephant was obliged to be on the stage for about an hour, and Ned was rather too fond of the bottle to remain so long without 'wetting his whistle,' so he sets his wits to work to find way to carry a 'we[e] drop' with him. The eyes of the elephant being made of two porter bottles with the necks in, Ned conceived the brilliant idea of filling them with 'good stuff.' This he fully carried out, and elated with his success, willingly undertook to play the fore-legs again.
Night came...Ned and the hind legs marched upon the stage. The elephant was greeted with round upon round of applause. ... The play proceeded, the elephant was marched round and round the stage. The fore-legs got dry, and withdrew one of the corks, treated the hind-legs, and drank the health of the audience in a bumper of genuine elephant eye whiskey, a brand, by the way, till then unknown. On went the play, and on went Ned drinking. The concluding march was to be made — the signal was given and the fore-legs staggered towards the front of the stage. The conductor pulled the ear of the elephant to the right — the fore-legs staggered to the left. The foot lights obstructed his way — he raised his feet and stepped — plumb into the orchestra! Down went the fore-legs on the leader's fiddle — over of course turned the elephant, sending the Prince, closely followed by the hind-legs into the middle of the pit. The manager stood horror struck — the Prince and hind-legs lay confounded — the boxes in convulsions, the actors choking with laughter. Ned casting one look, a strange blending of drunkenness, grief and laughter at the scene, fled hastily out of the Theatre, closely followed by the leader, with the wreck of his fiddle, performing various cut and thrust motions in the air....
...everybody holding their sides, music, actors, pit, gallery and boxes, rushed from the Theatre, shrieking between every breath,
"HAVE YOU SEEN THE ELEPHANT!!"
Hammerstein II, of course, if queried about the origin of his expression, might not have been inclined to mention the long, complicated and somewhat off-color history of his choice. Instead, an evasive, even if partially true, explanation, given without reason for the choice of 'elephant' ('giraffe', being only two syllables, was never in the running), might have seemed easier as well as less likely to provoke a backlash from the temperance set.
If the joke has to be explained, after all, it ceases to be funny. Everybody familiar with the details of the theatrical history would already understand the humor, and those who were not already familiar with the pun would not need to be enlightened.
To be plain:
I did not find "high as an elephant's eye" in print prior to Hammerstein's composition. 'Big', 'tall' or more rarely 'high' as an elephant was, however, a common hyperbole, also sometimes an accurate descriptive, in the 1940s, perhaps deriving from the widespread renown of Barnum's Jumbo (long dead by the 1940s).
I did not find the 'corn was high as an elephant', or variants, including '...elephant eye', in print prior to Hammerstein's composition. A common phrase at the time was, however, 'corn high as a horse's back'. Noteworthy, however, is that 'corn' was at the time in widespread use with the colloquial meaning of "old-fashioned or inferior music" (OED, paywalled). That sense along with the sense of 'corny' I mentioned earlier, derived from circumstances described in a 1946 American Speech article (21 234/1; as cited by OED):
M. Sandoz...The seed catalog [from c 1890 to 1910]...featured a great variety of seed corn...interspersed with short jokes and riddles, sometimes even cartoons. The jokes were all time-worn and over-obvious and were called corn catalog jokes or corn jokes, and any quip or joke of that nature was called corny.
Hammerstein did not invent "high as an elephant's eye" from whole cloth. He at least recycled scraps of earlier stock phrases, including 'corn as high as X', where X was commonly "horse's back", and also including 'big as an elephant'.
One origin story explaining the source of the line, although described as a "local legend" in The Philadelphia Inquirer (Philadelphia, Pennsylvania) of 18 May 1989, p. 252 (paywalled), at least fits the facts to known circumstances:
Local legend has it that when lyricist Oscar Hammerstein wrote "the corn is as high as an elephant's eye" for the musical Oklahoma, he was looking at the elephant atop the Buckingham Township Republican headquarters looming over the six-foot-tall August corn.
The account in the 1989 Inquirer appears earlier in a slightly different form in the Inquirer of 1986 (p. 349, paywalled):
Joe Kenny, 78, the owner of Kenny's News Agency and Bookstore in Doylestown, said Hammerstein's view from the balcony included an elephant that once set atop the Buckingham Township Republican Headquarters at Durham Road and Route 202.
Kenny said that Hammerstein discarded the image of corn as high as a cow pony's eye for a line in the song; instead he used "The corn is an high as an elephant's eye." He may have been looking at that elephant over the top of the more than six-foot-tall August corn, Kenny said.
Hammerstein wrote the lyrics of "Oh, What a Beautiful Morning" at the Highland Farm, Doylestown, Bucks County, Pennsylvania, next to Buckingham Township and across the road from a corn field.
A report on a fire in The Central News (Perkasie, Pennsylvania), 11 Apr 1940, confirmed the existence of the elephant on top of the Bucks County Loyal Republican Club headquarters (approximately 1 mile from Highland Farm via contemporary roads):
In spite of a fire which damaged their headquarters...members of the Loyal Republican Club of Bucks County...turned out for the scheduled meeting....
...A large electric sign, that of an elephant, toppled thru the roof as it weakened from the fire.
Presumably, the elephant sign was replaced when the club headquarters was repaired.
In any case, elephants were not in short supply in the area at the time. Elephant Road ran at least from nearby Dublin (about 6 miles from Highland Farm) to the unincorporated community of Elephant (about 11 miles from the farm); an extension of Elephant Road from Dublin to Doylestown might well have been replaced by the later State Route 313 sometime after Hammerstein moved to Highland Farm.