Oh, I had this problem in my apartment as well. In fact, the thing I love most about my house is the no-squeak floors and the fact that I don't have to worry about annoying the neighbors.
It'll be tough to eliminate the squeak, that's what squeaky floors do. I put a throw rug down plus a mat and I went barefoot and the the floors still squeaked. I even tried squirting baby powder into the cracks in the floor around the squeaks. This is supposed to work if it's the floorboards squeaking but doesn't have much effect on squeaks emanating from the joists, as was my case.
I ended up just bundling up and jumping rope outside in the courtyard.
I know you rent but I did a lot things as a renter without permission. If it's only one spot and one squeak you're looking at you might try something like this: SqueakNoMore
It's basically a screw with a head you can snap off below the wood surface so nothing sticks out.