This bicorne is a stiff felt hat, unlike the bicorne in my earlier post it won't fold flat, its also can only be worn fore and aft.
What I call hard hats might be better called stiff felt hats, they are hats made of felt where the felt is soak in water then steamed and then pressed in a mould and or rolled into shape. Once formed the hat stays to its formed shape and becomes quite rigid. Top Hats, Bowler Hats are usually made this way.
Purchased from a medieval market, this style is much flattened compared to some.