Leganoth, you might want to consider using foam to start the padding process, mold it into a cup that your head would fit (while wearing a coif) in at the top of the helm. Then if you were so inclined, sew the padding over (or under, depending on how you look at it) the foam so that no one'll ever know its there but you.
I suspect Ulrich will want to try the same with his transitional helm as it is a bucket top too.