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, still with the Stetson. The real version of the Doctor wore the Stetson while he was safe inside the Teselecta. (DW: The Wedding of River Song)

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).

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.