The main functions of a hat are twofolds: to provide balance, and to wetherize your skull.
Balance wise, the rules are simple: the more clothes you have on, the thicker your silhouette, the wider the brim of the hat; the more formal the dress. And the more formal the dress, is the hat. If one has a baby face or a thin face, you should stick to a Fedora.
Secondly, Beaver fur is preferred over rabbit or some other lesser creature. Beaver is more supple, sheds rain better, and is prettier. The barsalino makes the best, with stetson a distant second. These hats are expensive but with proper care, they will last for a long while. Avoid being cheap on hats or shoes;it will always show.
The wide brim of hat always go with a round face. they are more difficult to find. Crown height should be proportional to your physical height.

Woven or Linen Caps

They are fine in nearly all other cases . They are almost  required if one is not wearing a jacket. But please, no logos or other visible writing on the hat.

Westernized Hats

A baseball cap is an awkward device for preventing the rain from creeping down your neck, or in stopping the sun from blasting the same location. However, wool beaver hats especially shed rain beautifully, and even retain their insulative qualities when saturated .

Furthermore, unless you are an employee of the organization, you should not, without compensation, wear caps emblazoned with a company logo. jokes or slogans should be avoided no matter what.

Finally, the art of wearing a gentleman`s hat is one that men should be familiar with. There was, after all, a time when a gentleman wouldn`t leave his home without one but this rather staid convention is truly behind us.