Originally Posted by
sewbizgirl
I would sew on another strip of the backing material to make it long enough, and then finish.
I agree with this.
Making the binding wider will not solve the problem. You need to get the backing as long as it needs to be. If you have any of the backing fabric left (how about a little cut off the sides?), sew that onto the back where you need it. If you have none, I would sew a strip of the binding fabric there as a place holder. You only need an inch. It will be on the back, and will look like the binding is wider at that end, but will not detract from the rest of the back.
You might need to undo a few quilting stitches, and redo them after the piece is sewn on, but you will feel better about it the rest of your life if you do it RIGHT.