You are looking for the fusible, non-woven lightweight interfacing. I did one with it, the quilt was very heavy and it needs to be machine quilted. Also expensive and time comsuming to iron all that on. However, you may want to try using starch instead. My second t shirt quilt using starch came out amazing. Starch HEAVILY before cutting. I soaked them in starch solution from the grocery store,, wrung out and in dryer til damp but not dry. Ironed (wrong side) dry and then cut. Worked perfectly. ANd there is a little give so if you have to squiggle things around to get them to line up, it works. Good luck!
Ellen