Stetson

A Stetson was a type of wide-brimmed, high-stacked hat. It was typically associated with the United States. Not only did the Eighth Doctor receive one from the eminent American, Samuel Clemens, (DWM: Bad Blood) but Craig Owens instantly gave the Eleventh Doctor a Stetson when he heard the Doctor was bound for America. (DW: Closing Time)

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 one by Craig Owens. (DW: Closing Time) He wore it to Calisto B, Vegas 12 and the Seventh Transept before having the Teselecta impersonate him. The real version of the Doctor wore the Stetson while he was safe inside the Teselecta, and the Teselecta created it's own stetson to impersonate The Doctor's. (DW: The Wedding of River Song)

In Utah, the disguised Teselecta (with the Doctor on board controlling it) met up with Amy and Rory. When Rory complimented him on the hat, he replied, "I wear a Stetson now. Stetsons are cool," before the Teselecta's stetson was shot of it's head by River Song. (DW: The Impossible Astronaut). The Doctor had the real stetson on when the Teselecta faked the Doctor's death at Lake Silencio.

In his eighth life, the Doctor visited the Lakota Sioux Nation and met General George Armstrong Custer. During this encounter, he was wearing a stetson given to him by Samuel Clemens. (DWM: Bad Blood)

Behind the Scenes
The type of hat called a Stetson is manuactured by the Stetson company of Saint Louis, Missouri.