I agree with a lot of quilters on this site that said leave the old backing and put on a new one and then I would turn
it under all along the edge and butt it up against the hand sewn binding on the back and machine stitch it down.
Unless you feel it's not a big deal to unstitch the hand sewn binding on the back and then re-stitch it to the new
backing. That would be my first choice. Then I would machine tack it.