Assassin’s Creed Liberation has a very specific item group in the game - hats. You run around in assassin’s outfit which by default includes a hat instead of a usual hood. You can get a hood later if you like things in the traditional way. There are several hats in AC Liberation that Aveline can equip.
Assassin’s Hat. Aveline wears this by default. Assassin’s Hat is not really good-looking but it’s interesting entry since its head reminds you of a beak which may be a link to Assassin’s Creed main symbol - an eagle.
Assassin’s Hood can be purchased and equipped at any tailors shop. The price you must pay to look like assassins usually do is 500 écu.
Smuggler’s Hat. You can equip Smuggler’s Hat after you’ve collected all of Jeanne’s Diary Pages. These Diary Pages can be found in any locations around New Orleans, The Bayou, and Chichen Itza. Wearing Smuggler’s Hat will entitle you to a 50% discount for ammunition (should you decide to buy some; in my experience looting is way more practical).
Traveler’s Hat can be equipped after collecting all 12 alligator eggs in The Bayou. Wearing Travelers Hat will protect you from alligator attacks. That is not really necessary because you will not have much reason to be near those alligators after you’ve collected all the eggs.
Alligator Hunter Hat is a reward that you can get on Uplay for 30 uplay units. Alligator Hunter Hat stops alligators from attacking Aveline.