Without sounding daft, does the lead fit all the way in? I've had problems with the plastic coverings on phono plugs preventing the plug fitting into recessed sockets, so ,such so that I doctored a cable to fit.
Otherwise it's likely a soldered joint has failed and you'll probably find that there is so much stuff squeezed into a car stereo it will be impossible to get at the joint to fix it.
If you gently wiggle the plug in the socket do you get any better sound.
Wiggling has no effect and I took the front of the stereo apart to examine the contacts on the jack plug - it definitely fits right in and contacts properly.
If you're right about the soldered joint then at least I have a workaround in that earthing the iPod through its charging cable gives me decent sound quality.
In fact come to think of it the sound from the aux input has always been mildly substandard - when I installed it I compared the same piece of music on CD and mp3 and the CD was much richer, so perhaps this is just the final stage in the failure of a joint that has always been suspect.
To be clear on your proposed root cause, are you suggesting that the joint in question is part of the circuit that connects the shield part of the audio cable to earth, and that I am able to re-establish a connection to earth via the charging cable?
It could be a ground loop problem, you could try a ground loop isolated cable, I picked one up the other month from amazon, a Kingston one for under £10. I had a problem with a really low volume, it could be a similar problem to yours, especially if plugging it into 12v corrects it as it then shares the same ground.