Multiple realities
(covers information from several alternate timelines)
Pike's service cap
A service cap (also known as a peaked cap) was a type of hat, designed with a peaking brim and a short visor. It was often worn in the military.
Starfleet officers, including Captain Christopher Pike, owned a Starfleet issued cap as part of their uniform accessories during the 2250s. (TOS: "The Cage", "The Menagerie, Part I")
Starfleet Academy recruits and instructors also wore caps in the alternate reality. (Star Trek; Star Trek Into Darkness) Dress uniforms also had a corresponding service cap. (Star Trek Into Darkness)