These hats need to be used as you do stereotypes. They're use labels a character, defines and limits them, but if you carefully construct the wearer to reveal many traits not found in the iconic wearer then you can subvert the stereotype. This way, you can use the iconic hat to create humour, sarcasm and even a political argument just by having the wearer be different from the expected.
Otherwise, these hats might be ones you need to steer clear of when writing. Just so your character doesn't fade away under the power of the hat.