OP, Using a piece of tire as a tire boot is overkill, and I'm not sure if it would work very well. A piece of Gorilla tape, or as djb said a dollar bill. makes a good improvised boot.

My son rode back home using a $5 bill as a tire boot. I taught him this trick about 20 years ago with a $1 bill. He said it was inflation I also carry a Park tire boot in my tool kit.