Stetson

A Stetson was a hat, typically found in the western United States.

When in Nevada in 1861, the Eleventh Doctor wore a stetson as an alternative to a fez. (NSA: The Good, the Bad and the Alien)

The Eleventh Doctor was given a stetson by Craig Owens. (DW: Closing Time) He wore it to Calisto B, Vegas 12, and the Seventh Transept before having the Teselecta impersonate him, still with the Stetson. The real version of the Doctor still wore a stetson while he was safe inside the Teselecta. (DW: The Wedding of River Song) The hat is likely still in the Doctor's possession. In Utah, the disguised Tesselecta (with the Doctor still on board) met up with Amy and Rory. When Rory complimented him on the hat, he replied,"I wear a Stetson now. Stetsons are cool" before it was shot off his head by River Song. (DW: The Impossible Astronaut). Since the hat that River shot was part of the Tesselecta's disguise, the original remained safely inside the ship with the Doctor.