OP here. The strange thing is that I originally got a UPS tracking #, and a day later UPS handed it over and the US Post Office gave it a new Post Office tracking #.
So it looks like the UPS Store just handed it over to UPS as normal, and the UPS decided to transfer it to the Post Office (immediately, locally near me).
The UPS Store probably doesn't even know about this. But if it happens normally these days, I guess I learned my lesson. No need to pay more for UPS anymore.