pert:
I don't think there's any way for female headers that work with the standard square male headers on your Nano to be much more low profile because you're limited by the length of the male pins. You can get Nano clones without the headers soldered. The Pro Mini also comes without headers soldered and that board could be a good choice to use on a finished project where you aren't frequently uploading sketches.
I probably should have mentioned that many of my NANOs are clones, and I had gotten so used to soldering on header pins myself, I forgot the fact that the non-clones usually come pre-soldered.