Prop parented but not 'fit to' head is not following the model when the head moves...

I am making an FK animation using the Genesis 2 Male Slapstick 'porkpie hat' prop. The hat is parented to but not 'fit to' the head of the G3M Toon Duckie character and the 3DU Toon Rabbit because using fit to completely distorts the geometry of the hat so as to make it unrecognizable of a hat. However, although parented to the head bone of both models when I move those models such as what should cause the hat to move with the head the hat is not following the movement of the head!
In contrast, a prop cane that is being held in the left hand (each was parented to the left hand bone & not autofit in anyway) of each character does move when the character moves in such a way to cause the left hand to move.
I do notice that the pork pie hat has a G2M rig attached to it while the cane has no rig. I think I need to remove the rig from the pork pie hat and it will follow movement of the character then. How do I do that?
Does anyone know the fix for this?
Thanks.
+++++
Nevermind - I had mistakenly parented the hats to the root of the character rather than the head bone like I thought. I converted the rigged figure that they were to props by Edit - Figure - Rigging - Convert Figure to Prop... ...although as it turns out that wasn't really needed.