The headphone connector has 3 connections. When inserted part way, the tip makes contact with the right channel. When fully inserted the tip makes contact with the left channel and the ring makes contact with the right channel.
It sounds like your left channel has a problem and it is NOT your connections. Pulling the plug out a little connect to the right channel, which is OK.
It could be a noisy component, like a transistor or resistor, or a bad driver IC.