Can't put rivets back in with a chain tool. It is definitely true that you never want to squeeze the rivet out completely. I myself learned this the hard way.

Contrary to what most people say, it is actually possible to get the rivet back in. In my experience, there is not much damage to the chain.

Find an anvil or solid steel surface; not even concrete will have the right restitution properties. Put the pieces together the way it would be normally, except of course for the rivet. If your fingers are small enough, hold the rivet upright just exactly above the hole - tweezers or chopsticks might also work. Then strike several medium blows with a steel hammer. It must be very carefully done. If it gets in, even if it's a little slanted, the rivet will eventually straighten itself out when you resort back to using the chain tool. And there you go.