Rock the look. Believe in yourself and don't try too hard...just let the hat sit comfortably.
We think we look weird in hats because we're not used to seeing them. But in fact everybody can look good in a brimmed hat. The right brim balances out a broad face, and enhances a narrow face.
Go to an outdoorsy shop, like Cotswold, and try on lots of diffferent brimmed hats in different sizes until you find a style that you like. Take a friend for a second opinion, if that would feel better.