As I mentioned in my question, The Winter's Tale only survives in the text included in the First Folio of 1623, so it cannot be compared to other versions that might have represented earlier stages in the development of the play. For this reason, scholars can only rely on eyewitness accounts of performances in the early seventeenth century.
The only surviving eyewitness account of a performance of The Winter's Tale was written by astrologer Simon Forman, who attended a performance at the Globe on 15 May 1611. The description is included in a manuscript known as The Bocke of Plaies and Notes therof per forman for Common Pollicie. The website Shakespeare Online provides a modernised version of Forman's description of that performance (Mabillard, Amanda. "Going to a Play in Shakespeare's London: Simon Forman's Diary". Shakespeare Online. 20 Aug. 2000):
Observe there how Leontes, the king of Sicilia, was overcome with jealousy of his wife with the king of Bohemia, his friend, that came to see him. How he contrived his death, and would have had his cupbearer to have poisoned him: who have the king of Bohemia waning thereof and fled with him to Bohemia.
Remember also how he sent to the oracle of Apollo, and the answer of Apollo — that she was guiltless and that the King was jealous, etc.; and how, except the child was found again that was lost, the King should die without issue. For the child was carried into Bohemia and there laid in a forest and brought up by a shepherd. The King of Bohemia's son married that wench. And how they fled into Sicilia to Leontes. The shepherd, having shown the letter of the nobleman by whom Leontes sent away that child and the jewels found about her, she was known to be Leontes' daughter, and was then sixteen years old.
Remember also the rogue that came in all tattered like Coll Pixie; how he feigned him sick and to have been robbed of all that he had. How he cozened the poor man of all his money. And, after, came to the sheep-shearing with a pedlar's pack and there cozened them again of all their money. How he changed apparel with the King of Bohemia's son, and then how he turned courtier, etc. Beware of trusting feigned beggars or fawning fellows.
What is striking about this description is that the statue scene is conspicuously absent. (Antigonus running off the stage, "pursued by a bear", arguably Shakespeare's most famous stage direction, is also missing.) The statue scene draws part of its force from the conviction in both the characters and the audience that Hermione is dead, whereas in other Shakespeare plays, the audience tends to be in the know (e.g. in Much Ado About Nothing, Claudio thinks Hero has died but the audience knows this is a trick; in The Tempest, Ferdinand believes his father has perished in the storm but the audience knows he is on the island). In addition, the illusion of a statue coming to life can also be very strong. Testimonies of later performances, e.g. in the 19th century with Helen Faucit as Hermione and Macready as Leontes, show that this scene can be very powerful (see Shakespeare: The Winter's Tale, edited by Kenneth Muir; Casebook Series, Macmillan, 1968, pages 48-51). For this reason, it is difficult to believe that Forman would not mention the statue scene if he had seen it. This is what led to speculation that the astrologer might have seen an earlier version with a different final scene. However, hard evidence for this theory is missing.
Christopher Hardman cites another argument for the existence of an earlier version:
It has sometimes been said that the appearance of Hermione to Antigonus was evidence of a version of the play in which the queen did indeed die, as no one in Shakespeare's day would have thought it possible to see the spirit of someone still alive. (Shakespeare: The Winter's Tale. Penguin Critical Studies. Penguin, 1988. Page 15.)
(In Shakespeare's main source, Robert Greene's Pandosto, Hermione dies and Leontes eventually commits suicide.) Hardman also points out that in a version without the statue scene, the entire last act would be very different from the version we have now.)