Is “covering your head” not specific enough?

One does not wear ones panama whilst crossing the Antarctic, nor would one expect to be admitted to the Queens garden party while sporting a bicycle helmet. A scull cap may be appropriate whilst attending a synagogue but a “beanie” is preferable when chopping firewood in the snow.

Unless, of course, you are EXTREMELY eccentric.

If I am attending her garden party, the Queen will damn well accept my helmet, and like it.

Every hat I wear makes me look like a moron, they just don’t suit me; I only wear them when I absolutely have to