Hat head

orange hat (no brand)

As the Oxford Dictionary states: a hat can be "worn for warmth, as a fashion item, or as part of a uniform". 

summer hats trying to hide a rain hat 
(L-R: no brand - Veritas - Noa Noa - H&M)

The second "function" applies best to me. The opposite of the first too as summer hats have to keep warmth away. No need to say (but I'm doing it anyway) that rain hats come in handy when tears from heaven are falling down upon earth.

Basically it comes down to this: no matter the weather - warm and sunny or cloudy and cold - you can leave your hat on (as Randy Newman once wrote).

P.S. To what kind of headgear do you take your hat off?