I didn't have to take anything apart at all.
All I did was take a small, thin piece of a napkin, fold it once in half to fit the size of the headphone jack (making sure the strip was long enough to pull out if necessary, taping a tiny piece to the top of my iPod), and stuck it in there at the same time that I put the jack in.
It's been working perfectly fine so far, all I have to do is change the piece of napkin every week or so, to make sure it doesn't wear down.
My original problem was that I could only hear out of the right side of my headphones. I eventually noticed that the bottom right hand portion of the screen had the ability to slightly push down, making a light "clicking" noise. I thought of it as nothing, because it hadn't affected the performance of my iPod.
But recently, the above problem occurred. So that was my resolution. If any other type of suggestions aren't working (i.e., cleaning the jack with rubbing alcohol or physically taking off the screen) then try this.
It may or may not work, BUT BE CAREFUL: if you make the piece of napkin too thick and too short, it may get lodged in the very tip of the inside of the jack. Don't blame me if it doesn't work.
Best of luck!!