I think the envelope or pillow turn method would work well in this case. You could start with a piece of new backing that is exactly the same size as your top. Lay the new backing PRINT SIDE DOWN on the top. Stitch an even seam allowance all the way around the quilt but leave a piece open (I usually chose the bottom center for this). Turn the quilt inside out and stitch the seam closed. Now you can either tie or SID to secure the new back. You can decide to add regular binding around the edge or leave it as is.