After reading this thread when it was started, I needed to make some hard boiled eggs. I remembered a method I had used many years ago, but for some reason stopped doing it.
Use a strong pin or tack (washed, of course) to make a hole in the wider end of the egg, where there is often a little air pocket. You don't want to push the pin in all the way, just through the shell. Then proceed as usual.
I got my pin out and tried it, and I am convinced that it really does make a difference. I had already tried using vinegar, and while I found it made the shell itself softer, it is the inside membrane that I have trouble getting off the egg without damaging the white, and the vinegar made no difference with that.