lighter weight hats such as those made from fine straws like Panama, baku, and paribuntal or light, supple, thin-bodied felts
a good fit - not too tight
wearing a visor instead of a hat
After a hat is removed from one's head, hair will relax, take in moisture from the air, and in a short while show little or no evidence that a hat was worn. I have heard, anecdotally, that baby powder can be sprinkled into one's hair after hat removal and when combed out, voila, hat hair is gone. (I have not tried this first-hand.)
Perhaps (I don't know for a fact though) a good 19th Century/early 20th Century barber took into consideration where on a head a hat would lay and cut a client's hair with the lessening of "hat hair" as an objective.
When one considers the small negative effect of hat hair against the positives of protection against sun-related skin problems, keeping dry in the rain, keeping warmer in the cold, a nappy fashion statement, a new image and the like -- hat hair is no concern at all.