What types of hats do you wear?
If you wear knit hats like a Beret then there would be plenty of room for hair that is set pretty on the top of your head.
I would probably leave some hair down framing the face since you have such pretty curls and it would go beautiful with your face shape.
If you put the round of the hat at the hairline where the directon of hair flow changes and let the front hair frame the face this could be a pretty look. As well as not to disturb the updo.