As others have noted, numerous indications this is post-war (probably 1930s), with the celluloid diamond being a dead giveaway along with the sweatband and stitching method.
The latter is tricky though - I've seen many examples of clearly period-made pieces with later sweatband replacements, possibly done by the veteran owner. Sweat is salty, and salt is corrosive, so the sweatband is often the first thing to wear out or disintegrate with age. But in this case, the sweat band looks contemporaneous to the rest of the construction. Almost certainly a veteran's cap made long after WWI.