I actually have an older answer dealing with this, although it was in response to a question about whether Marie Antoine Carême invented it specifically, so just bear that in mind:

A lot of different publications like to give Carême credit as the inventor of the modern-day chef's toque, but this most likely has to do with the propensity people have to attribute changes in dress to an important, influential individual who makes an alteration on a whim and is copied slavishly. You find popular cooking/restaurant history books stating that Carême invented the chef's toque by stuffing a piece of cardboard into his hat, but generally speaking, you find academic texts being much more circumspect - because we don't really know. The frontispiece to Carême's Le Maître d'hôtel (1822) shows the "former" and "modern" chefs in their uniforms standing next to each other, and it's quite clear that neither has a tall, stiffened toque. Where the modern cook's is stiffened is, perhaps, in the hanging top, where it takes a firmly round shape that is accentuated by the tassel. And the only academic work that addresses this, as far as I can find, states that chefs "did not adopt the considerably less flattering stiff high hat (toque) until later in the century". (Accounting for Taste: The Triumph of French Cuisine, by Patricia Parkhurst Ferguson, University of Chicago Press (2006), p. 64) I suspect that Carême's is actually the hat described in the Gastronomica article as being worn under "King Louis-Philippe (1747-1793)" as a piece of whalebone formed into a hoop would hold this shape very well. Please note that, in line with the sloppiness of other aspects of this article, the Louis-Philippe with this lifespan was never king; his son Louis-Philippe (1773-1850) became king in 1830 and reigned to 1848.

Further, in examining visual sources, there is little to no evidence of tall hats until close to the end of the century. Nineteenth century chefs seem to have generally worn a white toque similar to the one that Carême actually wore, cut as a large circle and pleated into a wide brim; like other white accessories, such as aprons, these seem to have become starched to some degree as a matter of course (starching would help to keep dirt, oils, sweat, food bits, etc. from sticking too well to the fabric, and also would keep them crisp after laundering). Eventually, the starching became done more heavily, with the hat ironed over a block or cushion to keep it standing. The very tall toques we see today, an exaggeration of this style, don't seem to have come about until around 1900.

There is an outlier - [a portrait of an African-American man](https://en.wikipedia.org/wiki/File:Hercules_(chef).jpg) by Gilbert Stuart, which is presumed to be Hercules, George Washington's enslaved chef. This attribution is most likely made because his dress resembles a modern chef's: his coat and waistcoat are white, there is a white stock on his neck which somewhat resembles a handkerchief, and he has a tall toque; then it comes around again and feeds into the idea that the toque is this old. But we don't know that this is Hercules in the first place, and his headdress differs strongly from the traditional chef's toque in that it's open at the top. There's simply not enough information to interpret this portrait as a representation of late eighteenth-century chefs' dress.

There is a lot of speculation, hearsay, and pop history surrounding the chef's toque, but nobody has actually studied its history in any kind of scholarly way, so we can't say what was the exact impetus for the shaping was.