If the outer blocks look stretched, I might try placing machine basting stitches in the seam allowance. Pull the stitches up and secure the ends, then distribute the gathers evenly across all the blocks.
I would also spray starch both top and backing, maybe several times, allowing the starch to dry between applications. Spray starch will help stabilize the blocks so they don't stretch anymore. You could even "block" the edges with spray starch. Sharon Schamber has some videos on Youtube that demonstrate how to "block" blocks; it might also help with the quilt sandwich.
I think your best bet is to try to stabilize the edges to the size you want.
I don't know of a pattern that will "quilt out" excess fabric. When you start quilting in the middle, excess fabric gets pushed out to the edges. The stretching at the edges may not be due entirely to quilting; it could alo be that the edges have been handled a lot. Just handling fabric (especially bias edges) can cause it to stretch.