61st Regiment in Egypt , 1801, copy by Percy Sumner after Private Porter
(Anne S.K. Brown Military Collection, Brown University Library, photo courtesy of René Chartrand)
These round hats weren't replaced until around 1812, with gradually declining use since their inception in 1791, unoffciialy they were often adorned with plumes, fur crests, white bands, etc. Despite the regulations only allowimg for a cockade with the round hat. (Regiments serving in the tropics). Some examples were white which was intended to be more cool but obviously didn't do much in that regard. In addition to these hats, troops serving in the tropics were permitted to wear white Gaiters.
The replacement was a white belgic or "bang-up" shako initially in white, later in just in black felt and some even made their way to Europe and Canada.
http://www.warof1812.ca/tropicalshakos.htmFor more info on those white shakos and the specific regulations.