The hood on the chestpiece clips through the back slot (see screenshot).
tbh I always found it weird that the chestpiece had a hood while you already wore a hood. Would be neat if there was a way to make it invisible or whatever.
Honestly hoods hate every backpack. Much of that could be fixed if they disabled the jigglebones on the hoods when they had a back slot equipped. Like I use the white garb with the book when I’m running a little flap of hood sticks out. Oh well.