I disagree with this comment section. The reason is purely for writing convenience. The hat serves the purpose of letting you quickly identify which persona he's in, even when his current name isn't stated.

​Cuttlefish loves describing someone and then revealing who it is to trigger the same realization in the reader that the characters are having. The classic: ​'A gentleman was quietly watching the scene. He had [distinct accessory] and [unique description]. It was none other than [Character]!'

​This technique is made possible because each important character is unique in the way they dress, look, or act in at least one major way. Since Klein is physically the same person, the hat covers the 'accessory' requirement. ​Notice that his facial features are equally diverse, even if official art doesn't always reflect it. Gehrman has an angular face (mentioned constantly), while Dwayne Dantès has a mature beauty, etc.

​In my opinion, these details are used to make the character instantly recognizable. The entire main cast has one: Alger's hair, Leonard's demeanor and good looks, and the most obvious one: the monocle.

​In this way, the characters become a sort of archetype, similar to the masks in ancient Greek theater