I bought a J. Jill dress which had a lower V neckline than I wanted, but lo, there in my stash was a piece of black linen almost exactly like the dress.
You can see that the weave is looser, but it really doesn't seem to matter - it blends in very well. And that fringed edge? It's just the selvage, so I didn't have to hem that side of it! The most difficult part of the process was making sure it was in there straight.
I hand sewed it to the narrow edging with a backstitch, attaching it on the wrong side, and then pinked the raw edges. I intend to wash it gently in the washer and hang to dry, and hope it doesn't fray. It looks fine, like it belongs there!