Mercury/Hermes from Graeco-Roman myth also has a hat like this. It may be partly this dress choice (along with the common interest in poetry) that led the Romans and Germanic peoples to identify Woden/Odin/Wotan with Mercury, on the face of it a rather odd match.
Since Mat Cauthon from the Wheel of Time is an Odin Expy, he gains a really nice wide-brimmed hat in The Shadow Rising that he wears constantly even in battle.
Similarly, Gandalf from Tolkien'sworks had his appearance directly based off of Odin, complete with the iconic traveler's hat. This is where the popular image of a wizard wearing robes and having a wide-brimmed pointy hat come from.
Also in the wide-brimmed-hat club is Ankou, the Breton version of The Grim Reaper. Doesn't get much more awesome than that◊.
Santa Claus' hat is basically synonymous with Christmas. His elves are usually depicted with neat hats too.
The Japanese tale Hachikazuki-hime ("the princess with a big straw hat") gives us Fallen Princess Hachikazuki, who is told by her dying mother to always wear a big straw hat to protect her Memento MacGuffin, placed inside a small box hidden under it. After Hachikazuki runs away from home to escape from her Wicked Stepmother, the Nice Hat actually saves her life when she falls into a river but a fisherman spots the floating hat and pulls on it, also pulling the half-drowned girl underneath. She always keeps it on when in public, no matter what people say, honoring her mom's last request — even when working as a maid for a noble family, whose son falls for her... The parents of her beloved then demand her to do so, seeking to learn who she actually is before revoking their Parental Marriage Veto. The Memento MacGuffin is discovered under the hat, its contents prove Hachikazuki's noble heritage, and she earns her happy ending.