You'll need to collect at least three of the below hats and then equip them on any given mission to unlock this achievement. Please note that you'll need to equip three different hats, you cannot use the same hat twice.
Name: Ace Hat
Owner: Royalist Buck (Ace)
Location: The Royal Impound, An Audience With the Queen.
Name: Air Force Helmet
Owner: Royalist Grunt
Location: Royalist Tax Enforcers, No Stone Unturned, On Her Majesty's Secret Birdie, Royal Logistics, Royal Impound, A Hustle Worth the Tussle, A Lonesome Stranger, Clean House, Robo a Robo, Through the Royal Palace, An Audience With the Queen.
Name: Beret
Owner: Bandit Thug
Location: Wuthering Heights, Explosives Transport, Lone Ranger, Turret Plant, Tick Tock.
Name: Cavalry Riot Helmet
Owner: Royalist Dragoon Grunt
Location: A Hustle Worth the Tussle, Brawl Club, Clean House, Prison Break, Through the Royal Palace, An Audience With the Queen.
Name: Ce Petit Haute
Shop: Le Chapeau de Licorne
Water Cost: 2200
Rep Cost: ---
Name: Copperback's Cap
Owner: Royalist Copperback (Copperback)
Location: Prison Break, An Audience With the Queen.
Name: Heavy Chopper
Owner: Heavy Chopper
Location: Safe and Secure, Big Rig, Tick Tock, Chop's Shop, Top of the Heap.
Name: Military Officer's Cap
Owner: Royalist Officer
Location: The Royal Impound, Clean House, Robo a Robo, Through the Royal Palace.
Name: Scrapper Captain Hat
Owner: Bandit Goon (Jim-Bob)
Location: Beat the Brute
Credit to Woudo from the Steam Community for the information.