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.