Have you got a simple design of a satin stich circle or other fitting shape? Stitch that on your (still hooped) design. Cut it out and stitch that on your quilt - having the neat edge may make it easier to machine stitch the patch to the quilt.
I'd sure love to do it on the machine - my hand stitching isn't so beautiful anymore LOL
Jan, I would unhoop it and then get you some HB (wonderunder) and iron it on. then you could use a whip stitch to make it look like you hand stitch it there.
What a great name 'wonderunder' it called bondaweb in the UK. Not half as exotic