Open the Equipment/Loadout menu before starting the contract
Select the headgear item you want to remove
Unequip it or replace it with another item
If already in a contract, use the truck loadout screen if available
If the item is on your character, press the inventory/equipment key and switch it off
On VR, use the grab/menu interaction to remove or swap the headgear
If it is stuck, leave the contract and change your loadout in the main menu