Quote Originally Posted by Matroskin Kot

If you need to be general, I would just say 'hat', which basically covers any non-specific type of head covering.
For example:

"Я никуда не иду без головного убора."
"I don't go anywhere without a hat."
Can I say, "I don't go anywhere with my hair uncovered"?

And, by the way, Matroskin Kot, if you don't mind, the Russian phrase I would rather say differently, "Я никуда не хожу беэ головного убора." It just sounds more natural to my Russian ear...