There's no reason why you can't use the rest of the t-shirt in a quilt. Think about why you want it in a quilt in the first place. You want the design that's on the front and/or the back. The rest of the t-shirt is just fabric. It could be used for sashing, or small blocks. It would need to be stabilized in the same way as the parts with the designs.

I've thought of using iron-on designs to transfer a design to parts of the t-shirt that wasn't otherwise being used in the t-shirt quilt. But our t-shirts are wearing out, and that's why they're available to go into a quilt, so finding decent fabric elsewhere in the t-shirt can be difficult.

I've made several t-shirt quilts, but I always use a stabilizer to keep the blocks from stretching out of shape. Since I cut the shirts based on the size of the design on each one as opposed to cutting them all the same size, I use extra blocks to fill empty spaces between my t-shirt blocks. I wouldn't have any problem using the back side of a shirt if I needed another filler block. I would treat it with a stabilizer just like the main parts.

I have made several t-shirt quilts without stabilizer and added cotton sashing between them , I would have the t-shirt fabric facing down and the cotton sashing on top as I sewed it. I tied these so I'm not sure they would be able to long arm them without the stabilizer tho. I also used different sized t-shirts and the pockets also.