If you go to the Lodestone and look under the Play Guide header there is the Eorzea Database. You can search up any dungeon or 24-man raid in the game and see exactly what can drop from each individual treasure chest.
In the case of Dun Scaith, all of the hats have a chance to drop from the chests following the first two bosses: Deathgaze and Ferdiad. So now you know that if you beat Ferdiad and it hasn't dropped, it's not going to that run. I don't suggest you use this to quit out of runs that don't drop the hat after the second boss but now you know not to expect it drop from the last two bosses.