Due to its lack of special ability or AC benefit, the fedora is often ditched at the beginning of the game. (It also has no value when dropped in a shop.)
Fedoras are made of felt, with a wide brim (about 2.5 inches or 6.4 cm) and a distinctive lengthwise crease in the crown. The style goes back as far as the 1880s and got its name from the French play Fedora, in which actress Sarah Bernhardt, playing the title character "Princess Fedora", wore one. They were especially popular from the 1920s to the early 1950s, the time period in which the Indiana Jones series is set (see above). Fedoras fell out of widespread fashion after the mid-1950s, but are still occasionally seen worn as accessories.
Some hats can only be worn if you're willing to be jaunty, to set
them at an angle and to walk beneath them with a spring in your
stride as if you're only a step away from dancing. They demand a
lot of you.
[Anansi Boys, by Neil Gaiman]