Overflow gaskets are usually wedge-shaped...thicker on one side, tapering to thinner. This is because the back of the tub is often slanted. Check that the orientation of the gasket it such that there is approximately even squeeze at top and bottom as you pull in the screws.
At this point, I would REPLACE the gasket as part of your inspection process. Inspect the overflow to ensure it is plumb, and in the horizontal plane is parallel to the back side of the tub.
Silicone sealer doesn't hurt, but may not solve a serious installation error or wrong gasket.
If the drain looks like the one below, you may need to pull the "shoe" out with a wire coat hanger.
I snake from the upper overflow, but the linkage and the stopper need to be pulled out first.