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The soft cover roll sticks to your skin, and the stronger tape sticks to the cover roll tape. This is a standard technique in McConnell taping, but it can work for KT tape too. It's nice because if the KT tape falls off, it won't automatically take the cover tape with it, depending on where you have the tape.
Ex of cover roll tape:
https://www.amazon.com/gp/product/B000GCRWNA/ and it works great, thought if you reapply it if you messed up the first time, it will start to peel off, but not if you apply it the first time. I bumped it a few times and this stopped it from coming off.
Is it just the blade/heel of your hand that gets scuffed up while centering? What about covering just that area with medical tape? If you put on a piece at a good angle and length so the edges didn't get caught, I'd imagine it would hold up well and leave the rest of your hand free.
This type stuff: https://www.amazon.com/Cover-Roll-Stretch-10-yards-Hypoallergenic/dp/B000GCRWNA
Heck, even a strip of duct tape could do the trick.