I went to my LBS and talk with the owner. I explained my situation and told him I was trying to find an option for that palm/thumb.

He said he could put the gel tape under my bar tape but it would not really solve my problem and those gel tapes are not as "soft" as you'd think. And it would also thickens oo much the hood, etc.

When I told him about the soft ice gel pack he told me it would be the best solution, as long as they were not too bulky. I checked what I have home and they are 4x6''. Too big for my bar. So I'll see if I can get smaller one at drugstore tomorrow. If not, some sort of facecloth wrapped around (even if it means I have to cut it to make it fit) will have to be it.