(covers information from several alternate timelines)
Jonathan Archer wearing a Starfleet baseball cap
A baseball cap was a type of cap originally popularized on Earth from its association with the sport of baseball.
A woman wearing a MAGA hat
Preceding the Second Civil War in the mid-21st century, protesters who supported Donald Trump often wore red "MAGA" ("Make America Great Again") hats. While warning the Kiley leaders about the perils of social division in 2259, Captain Christopher Pike showed them historical footage of such people trying to storm the United States Capitol. (SNW: "Strange New Worlds")
Baseball caps were part of standard United Earth Starfleet attire during the 22nd century. Captain Jonathan Archer and Crewman Alex had such a cap with their ship's registry, "NX-01", stitched on the front. (ENT: "Broken Bow")
Archer, as well as Travis Mayweather and Crewman Ethan Novakovich wore these caps while exploring the first M-Class planet the crew of Enterprise surveyed. (ENT: "Strange New World")
Reed and Tucker wore caps while stranded on a shuttlepod. (ENT: "Shuttlepod One")
Archer and Trip Tucker wore these caps while crossing the desert terrain on the Torothan homeworld. (ENT: "Desert Crossing")
T'Pol wore this cap while trying to be unseen by MACOs after she was relieved of duty by Archer and was confined to her quarters. (ENT: "Hatchery")
Archer also wore his cap during his venture into Vulcan's Forge to help protect his eyes from the Vulcan sun. (ENT: "The Forge")
Tribbles mascot wearing a backwards baseball cap
The mascot for the Tribbles commercial was a cartoon tribble with a face that rode on a skateboard and wore a backward baseball cap. (ST: "The Trouble with Edward")
Baseball caps were worn by the Niners baseball team consisting primarily of the Deep Space 9 senior staff. The cap featured a baseball with the horizontal design of DS9 overlayed on the right-hand side stitching. Their opponents, the Logicians – from the all-Vulcan crew of the USS T'Kumbra – wore a cap with an IDIC, the symbol of their people, on the front. (DS9: "Take Me Out to the Holosuite")
During practice sessions leading up to their game, some of the Niners wore plain caps in various colors, while Captain Sisko wore the cap of the Homestead Grays, a team that played in the Negro National League from near Pittsburgh, PA, and Jake the Atlanta Braves.
In 2381, there was a baseball cap on display as part of the late Kerner Hauze's collection. (LD: "Kayshon, His Eyes Open")