Tapabord

I am pretty sure this cap was worn by Canadians during the Seven Years War. It certainly was in use in the 1690s
This reconstruction by F Back is of 1750s Miliciens.

(Also see Karpus). 'English style cap, sometimes called Bourguignote by sailors. It is a day and night cap, with flaps that can be turned down for protection against the wind and sand' Furetiere 1690.

For references to its use in New France go here to an excellent book by Timothy Kent.