A quick run-down of some of the common styles...
Soft felt hat with a wide brim, as seen on everyone from Bogart to Bond. My personal favourite.
Felt (or straw) hat with a flattened top and a short brim. Can be short-crowned version of a trilby in Britain
Stiff felt hat with a single centre dent in the crown and a brim fixed in a tight upwards curl.
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