Stetson

The Doctor wears a Stetson.

A Stetson was a hat, typically found in the western United States. The 1,103-year-old Eleventh Doctor donned a Stetson given to him by Craig Owens, (DW: Closing Time) while in Utah. When Rory complimented him on it, he replied, "It's a Stetson. I wear a Stetson now. Stetsons are cool" and the Stetson is promptly shot off his head by River Song, who had appeared behind him, echoing a previous incident with a fez. The current wearabouts of the stetson are unknown. it was probably blown into the desert by the gun. (DW: The Impossible Astronaut)

In the stetson's own timeline, it is given to the Doctor by Craig in episode 12, and the Doctor then wears it through episodes 13 and 1.