Sounds like you did it right. Hot pan (not super hot, just medium), hot fat then add the egg and let it cook undisturbed for a minute or two before you try to move it. The pan must be clean as a whistle before you start. Any residue will serve as a sticking point.
You may have to gently dislodge it from a spot or two but it should be relatively easy.
All that being said, while 95% of my cookware is tri-ply SS, I use a cheap teflon on aluminum skillet to do my eggs. No thought involved-easy peasy.
"If you want to make an apple pie from scratch, you must first create the universe." -Carl Sagan