Go for a Fedora. Its the kind of hat worn by Indiana Jones and Humphrey Bogart in the last scene of Casablanca. It right for any time from 1920 to 1960. Furthermore, you can wear one and be any number of things - newspaper reporter, private investigator, Russian spy, businessman or International Man of Mystery. Go for it!
Cloche hats were the felt bell-shaped hats popular in the 1920's. These were fashionable for women. Cloche hats were invented in 1908 by Caroline Reboux. These hats made a comeback in the 1960's.
Obviously, hats! Scarfs, NO WAY!
Baseball, football, basketball and hockey were popular in the 1950s.
Christmas hats, or sleeping hats?
The Twist!
DIAMONDS, FEATHERS, STUDS ARE ALL POPULAR ON WESTERN HATS
Yes, men wore hats in the 1920s. Hats have been popular for hundreds of years.
Panama hats
In The 1950s
Gunsmoke
Audrey hepburn
Washington Senators in the 1950s (specifically, Jim Lemon 1B)--(from dahuppster)