Remember this thing?
Nothing wrong with it per se. There is plenty of evidence for floppy elf caps. Eighteenth century soldiers wore these to preserve their expensive cocked hats when it was time for manual labor. For instance, there are these hearty chaps toiling away on the defense of Boston in 1775:
The Packet, Mark Tulley argues this is the most common cap, and I agree.
Were there other types of fatigue caps? Probably. Like enlisted men's coats, the caps tended not to survive. My friend Jason Melius, assembled some intriguing evidence for another kind. Take a look at some details in the works of Paul Sandby.