Here is a short list of hat styles I would love.
Balmoral Bonnet
https://upload.wikimedia.org/wikipedia/commons/1/16/Balmoral_bonnet_black.jpg
Boater
https://upload.wikimedia.org/wikipedia/commons/c/c5/BoaterStrawHat_wb.jpg
Boater
https://upload.wikimedia.org/wikipedia/commons/1/1c/Bowler_Hat_sw_fcm.jpg
Cloche
https://upload.wikimedia.org/wikipedia/commons/1/13/Vilmabanky.jpg
Sombrero Cordobés
https://upload.wikimedia.org/wikipedia/commons/a/ad/Sombrero-cordobes.jpg
Coonskin
https://upload.wikimedia.org/wikipedia/commons/4/43/Cap_of_American_opossum_with_a_raccoon_tail_%28Davy_Crockett_style%29.jpg
Dixie Cup Hat
https://upload.wikimedia.org/wikipedia/commons/8/8d/US_Navy_080923-N-4965F-009_Navy_Region_Hawaii_Ceremonial_Guard_forms_up_in_preparation_of_a_burial.jpg
Fascinator
https://upload.wikimedia.org/wikipedia/commons/thumb/9/96/Chloe_Moo_of_Darwin_-_2013_Myer_Fashions_on_the_Field_%2810705418974%29.jpg/1920px-Chloe_Moo_of_Darwin_-_2013_Myer_Fashions_on_the_Field_%2810705418974%29.jpg
Fez
https://upload.wikimedia.org/wikipedia/commons/7/78/Fes.jpg
Glengarry Bonnet
https://upload.wikimedia.org/wikipedia/commons/thumb/c/cd/5_SCOTS_Soldier_with_Bayonet_Fixed_on_Parade_in_Dumbarton%2C_Scotland_MOD_45152907.jpg/1920px-5_SCOTS_Soldier_with_Bayonet_Fixed_on_Parade_in_Dumbarton%2C_Scotland_MOD_45152907.jpg
Panama Hat
https://upload.wikimedia.org/wikipedia/commons/a/a1/Panama_hat.jpg
Pillbox Hat
https://upload.wikimedia.org/wikipedia/commons/a/a2/DorisDay-midnightlace.jpg
Pith Helmets
https://upload.wikimedia.org/wikipedia/commons/d/d3/PithHelmetTruman.jpg
Rastacap
https://upload.wikimedia.org/wikipedia/commons/8/8a/Rasta_Man_Barbados.jpg
Toque
https://upload.wikimedia.org/wikipedia/commons/3/3f/Chef_Hat.JPG
Top Hat
https://upload.wikimedia.org/wikipedia/commons/thumb/8/84/Austin_Lane_Crothers%2C_photograph_of_head_with_top_hat.jpg/420px-Austin_Lane_Crothers%2C_photograph_of_head_with_top_hat.jpg