Spraying contact cleaner into the socket is mostly serving to coat everything else in the cavity. And sandpaper is harsh enough to remove any plating that might be on the socket.You could spray some contact cleaner on the sleeve of the cable jack and then plug it in/out of the socket a few times, spin it around as well. Or use some fine sandpaper wrapped around a pencil and spin that around inside the socket a few times to clean off any oxidation.
If a soft pencil eraser isn't sufficient to remove any corrosion or gunk, spray some contact cleaner on a rag or Q-tip and wipe the surface with that. If that doesn't fix the problem, the problem is something else.
Sometimes those sockets just wear out and have to be replaced. They're cheap and easy to swap out so don't spend a lot of time trying to fix this one.