Tub overflow drain upstairs with water damage on downstairs living room ceiling
here's a typical tube/over flow drain set up:

The black ring at the top of the picture is a rubber gasket. It forms a seal between the over flow drain pipe and the wall of your tub. If the tubs' drain or stopper needed maintenance the lever would be removed. This may have caused the rubber gasket to become awry. It's tricky to keep it in place while securing the screws that hold the cover.
That is the most likely location of your leak. To repair you must remove the two plate screws and verify the gasket is not split or damaged and centered in the opening. Further the pipe behind it must also be precisely centered to the opening and gasket. Start one screw first turning it until it stays in place. Tighten the next one snugly and finish tightening the first too.