Texture settings are completely separate from figure morph settings — whatever you do with one can never have any effect on the other — so it doesn’t matter. Just transfer the figure object, then make a copy of the skin texture and edit it to add the bits you want. It’s called “second skin”, and its only drawback is that you can’t change the actual shape of the object.
Unless… does “skin an extra piece” have a specific meaning in 3DS MAX related to actual object geometry instead of texture editing? Different 3D programs use different terms for the same thing (and the same terms for different things), and this can cause confusion.