Military hats for Star Fleet officers? What? I understand that this is a "reboot" and a new timeline as of some 25 years from the time of the movie but millitary hats?
A homage to Pike's hat from "The Cage", which ended up being used as set dressing rather than as an item of wardrobe. It was originally created as Robert April's officer cap, and was mentioned in an earlier version of the script of "The Cage.":