Looks like the size and shape of some of the 18th century bonnets, which the military revived around WWI for their khaki drab (olive drab) bonnets.
I had one of those and it was pretty good at keeping off the sun, more practical for California than the small Balmoral bonnets.
Proud Mountaineer from the Highlands of West Virginia; son of the Revolution and Civil War; first Europeans on the Guyandotte