Well, I suppose you could add a layer of foundation fabric underneath, like muslin? I've never tried this, but an extra layer of fabric between it and the batting would probably help. I would use spray or glue basting to hold it together as you sew.
If it was my project, I would square it up, add another border to make it larger (maybe six inches all the way around), then sandwich it with batting and backing and quilt it with a criscross diamond pattern. You should be able to get a walking foot for the quilting for any sewing machine you have - check in your manual for the correct foot. A lot of sewing machines come with them already.