Stetson

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

The Eleventh Doctor was given a stetson by Craig Owens. (DW: Closing Time) The Teselecta wore it while impersonating the Doctor in Utah. (DW: The Impossible Astronaut, The Wedding of River Song)

The Doctor later stated, "I wear a Stetson now. Stetsons are cool" before it was shot off his head by River Song. (DW: The Impossible Astronaut). However, the real version of the Doctor still wore a stetson while he was safe inside the Teselecta. (DW: The Wedding of River Song)