This year I’m not buying paper wrap any more. I’ll use up what I have, but I’m switching to cloth wrappings. In Japan, wrapping gifts in cloths is called Furoshiki, and there are many ways to fold a square of cloth to carry almost anything, it’s quite cool.
I’ve made some bags with ribbons sewn into the side seams, as well as squares of fabric with overlocked edges.
I do like this idea, but your gift would have to be the exact size for it to work (would be nice for a book or something boxed).