I am also learning that modifier, it is also relevant if the normals are oriented outwards or inwards. it is best to deform it manually, you will also have more control of the final result
from what I see here the hat deforms where the headphone rests and then slightly covers the edge, the headphone appears to deform a little in the area where it rests with the head
Cull faces -> Front
Watch your offset distance… in my setup which is to real world units for a character it would be probably something like .01 (it might be signed, I can’t remember atop of my melon, haha)
can you upload your blend file? I could have a look at it and make it work, perhaps. Usually the Shrinkwrap modifier works like a charm in situations like this
you can also use the Mesh Deform modifier, create a cube around the area you want to deform, subdivide it a bit, give your headphone a Mesh Deform modifier with the cube as Object, move some vertices of the cube to deform the headphone. It has the advantage of being a non-destructive method