I learned something interesting about flat hat brims while talking to hatmakers about a special hat I was looking for. If you happen to want to keep a brim really flat, ala the Wyatt Earp hat in movie Tombstone, its important that the crown be the correct shape to fit your head. If you have a long oval shaped head, like I do, and put on a hat that's shaped to fit an oval or round head, the front and the back of the brim will droop. No amount of brim pressing will correct that
That's why to get a real custom fit, the best hatmakers measure the shape of your head, along with the size, with a device that looks like a medieval torture hat. Then they block the hat form to the dozens of measurement it takes, so the hat will conform precisely to the shape of your head without stressing the brim to cause it to deflect or buckle.
And learning that fact solved the mystery of why most stock hats, at least the stiff ones, don't fit me well, even when the tags say they're the correct size. Pulled down tight against the front and back of my head there's sometimes a 1/4" of airspace on the sides where it doesn't touch my head at all.