Why not just put the new shirt into a pillowcase used to store the quilt or used as a pillow with the quilt and forget all the extra work?
I realize it might be difficult, but the person wasn't helping out by leaving it this late in the game. You don't want to reward their delay - do you?
Just a thought. I've been caught the same way before and it is not fun to have to stop everything and try to regroup.