Seen on Dragnet, Sgt. Friday's hat was a signature of his costume. The wardrobe department had multiple copies of the same hat he wore during the show.
This sketch was made after the hat he was wearing on the sequence we were filming was declared missing and possibly misplaced. The whole crew went searching for it. Later It was discovered that Jack had left it in his car just outside the studio.
In the 1920's to the 1940's hats were worn as common fashion accessories of clothing for either genders.
Jack Webb wore many different hats (sort of speak) in the show as a producer, writer, director, and actor.