I'd use shortish hair, like Paris65 by Fabiana. That way, you have little to mess with. Push  your hat down until it is sitting right, and don't worry about the hair-pokethrough. With your morphing tool (brush):

Click on Create tab

Set it on push

Set magnitude to .02

Twirl your camera around until you are looking squarely at the hair poking through the hat.

On my new PC, the brush acts a lot faster, so tailor your settings to your system. You might want to set magnitude higher if it's all taking too long.

Be aware that where the crown meets the brim, excess pushing of hair is going to result in visible bald spots under that hat.

Hope this all sort-of makes sense. Two things to remember: 1 start the brush action with the highlight over the mesh you want to act on and 2. hold down the shift key to keep focus on that mesh (works for the most part)

Monterey/Mint21.x/Win10 - Blender3.x - PP11.3(cm) - Musescore3.6.2

Wir sind gewohnt, daß die Menschen verhöhnen was sie nicht verstehen
[it is clear that humans have contempt for that which they do not understand] 

Metaphor of Chooks