Looks to me like an attempt to quilt in sections - keeping part of the backing off initially, to reduce bulk. I was initially thinking - just piece the backing together and finish quilting. But the amount of damage to the batting might be the deciding factor. If the batting isn't too bad, and you are comfortable repairing it, that's what I'd do. Repair batting, piece backing, finish STID on the quilt. Then, I'd probably unstitch the top edge (or cut it) and bind all 4 sides.
But, if the batting is really bad (in the photo, it looks like it's completely crooked/half-missing?) I might feel obliged to take the 3 layer apart and start over. I would NOT however, modify the front/top of the quilt. That is done and unless you have major issues like seams unraveling I'd leave it as it was done by the aunt, even if it's not perfect.