Car overheating

On Friday, a little after starting out on an hour trip, the low engine coolant light came on. I was nowhere near a store so I just kept my eye on the temperature gauge. The temperature was fine. Even a little below where it usually is. When I got to my destination there was an emergency back home so I had to come straight home. There was no time for me to stop and wait for the engine to cool. The temperature was fine. Saturday I drove to Walmart, which is about 15 minutes away to buy coolant. Everything was fine. Sunday morning I added coolant. At that time I didn't see any in the reservoir. I drove about 20 minutes away and came back and everything was fine. Later that night I drove again about 20 minutes away and by the time I got there it was just about to be in the red. I had to stop a few times but I made it home. This morning I looked and there was just a little coolant in the bottom of the reservoir. I put cardboard underneath the car and added more coolant. I suppose there's a leak somewhere because there was some on the cardboard but what I don't understand is why would a car not overheat when the reservoir was empty but would then overheat when there was a little coolant in there? That just doesn't make sense to me.

I just checked the cardboard again and it's definitely leaking. Could me putting the coolant in yesterday have nothing to do with it overheating? I don't know too much about cars but I do try to understand why things happen. All I know is that I read the coolant travels back and forth between the radiator and the reservoir. Could it be that me putting coolant in, not only did absolutely nothing to help it because of the leak, but any coolant that was still in the system after the area of the leak was there until last night and that's why it overheated?