An update to my OP:
I got a flat, so I decided to try my original idea of putting an old inner tube into the tire before putting the good inner tube in there. I wasn't able to mount it back onto the rim because there wasn't enough room for both tubes in the tire. As I was trying to mount the tire, the tube closer to the opening of the tire kept popping out and wouldn't stay in. I tried hard to make it stay in, but it just wouldn't. So I had to give up on the idea.