Same as the achievement above, however once you’ve sent the hat through the teleporter to Xen you’ll need to continue carrying it to the Nihilanth fight. Once you have the hat you’ll need to painstakingly carry it from here to the end of Chapter 14: Lambda Core, and place it in the teleporter.
You can find the Hidden Hat during Chapter 3: Unforseen Consequences, a little bit into the chapter you’ll return to the front desk area from the start of the game, instead of going through the vent to your right, go back out towards where you got off the train on your right are some pipes you can jump on, the hat is hidden in the corner.