In the early 19th C. when the eldest daughter married, did the second oldest daughter become the "Miss Whatever," or did she continue to be identified as "Miss Whoever Whatever?'
"Miss [surname]" would be used to refer to an unmarried woman when it was unambiguous in the context.
To address something to "Miss [surname]" when there were several people of that name it could apply to would be incorrect, as a matter of practicality as much as etiquette, though if it did happen then it would likely have been assumed to refer to the senior person of that name in a household (not necessarily one of a set of sisters, if there were cousins or other relatives in the household of the same name) unless there were some reason to deduce otherwise (e.g. the stamp of someone known only to one of the women).
The eldest unmarried daughter (especially of the senior branch) could style herself "Miss [surname]" on a calling card, on signing a hotel register, and similar. This would indeed pass to the second eldest upon the eldest marrying, taking religious orders, or dying. I do not know if there was consensus on other reasons why one might leave a household. (Other reasons for leaving a household could be matters of scandal in themselves).
When a young lady has been formally introduced to society by her mother, she uses for her first year of calls, cards that bear her name below that of her mother. She assumes a private card only when she is no longer a débutante. The joint card, as it is called, should be larger in size than the card her mother ordinarily uses, and the young lady's Christian and middle names should be used unless she is the eldest daughter of the family. — Lillian Eichler, Book of Etiquette Volume 1
The wife who is the senior matron of the senior branch of a family may drop both her husband's first and middle names from her cards, and have them read simply: "Mrs. Robinson." Her eldest unmarried daughter is entitled to use a card reading: "Miss Robinson." When the name is a very ordinary one like Brown or Smith, it is always wiser to use the Christian names to avoid confusion. — ibid.
It is not considered dignified for a woman traveling alone to sign herself in the hotel register without the title of "Mrs." or "Miss." A married woman should register as "Mrs. Harris K. Jennings," an unmarried woman as "Miss Mildred Jennings." It is decidedly bad form to sign oneself "Millie Jennings," or "Flossie Jennings" for Florence. The full first and last name should be written out and preceded by the correct title of "Miss" or "Mrs." Only the eldest daughter, or only daughter, of a family may sign herself, "Miss Jennings." — Lillian Eichler, Book of Etiquette Volume 2
When announcing guests, the butler should ask, "What name, please?" not in the indifferent, sing-song manner so characteristic of butlers, but in a cordial, polite tone of voice, and with a genial smile. Having been given the names of the visitors, he announces them in clear, distinct tones. These announcements are made while the guests are entering the drawing-room. A mother and two daughters are announced as: "Mrs. Smith, the Misses Smith." If the given names of the young ladies are called, the form of announcement is: "Mrs. Smith, Miss Smith, Miss Alice Smith," the eldest daughter of a family being given the privilege to use the title "Miss Smith." In announcing a gentleman and his son, the butler says: "Mr. Blank, Mr. Francis Blank." — ibid.
(Eichler's book is an interesting case of the value of appropriate writing, she was a copywriter whose job was to advertise the earlier 19th century Encyclopædia of Etiquette by Emily Holt, which was no longer popular. Having succeeded Doubleday concluded that she was clearly a good writer, tasked her with producing a rewrite in more modern language and the result sold in the millions).