A couple of quick thoughts:
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-I have found that simply by wearing a tie in public, people tend to assume that I work wherever I happen to be. I regularly get questions in banks, stores, and restaurants that are intended for an employee or manager. Often I can answer them and try to do so in the most courteous way possible. To the extent that someone associates a hat with being "dressed up," it might enhance such an effect. It seems to me that merely being dressed more formally than other folks in a place of business will lead to you being perceived as part of the business.
Yeah, going to Grad School kind of messed with my melon and sometimes a cigar is just a cigar.
That being said, how we choose to appear in public says a lot about who we are and what message we are sending about ourselves. Sometimes, it even says GOOD THINGS about us.
Something to think about, or not...