A slouch hat was a type of hat where one side remained flat while the other side was pinned against the cap.
During the American Civil War, General Robert E. Lee wore a slouch hat as part of his military uniform. (TOS: "The Cage")
Harcourt Fenton Mudd wore a type of felt slouch hat when he was captured by the USS Enterprise in 2266. (TOS: "Mudd's Women")