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 was given to the Doctor by Craig. According to Craig, it's "From Sean's stag".