After decades of not being able to peel fresh hard-boiled eggs, I found a way that so far is almost fool proof. I don't wash the eggs until just before cooking. Then I wash the eggs and take a push pin and put a small hole into the large end trying not to go in far enough to break the membrane. Next I put them in a saucepan and cover with water then add a good glug-glug of vinegar. Put them on the fire and bring to a boil. When they really start boiling, turn the heat off and cover then time the eggs. About 20 minutes for real large eggs and about 10 for bantam. When the time is up I take them out of the hot water and put in a colander and run very cold water over them or in another pan and keep replenishing the cold water until the heat is out of them. When the eggs are cool I put them back in the original water and bring them to a boil again, this time only letting them sit in the hot water a minute or two. Then I put the pan in the sink and run cold water over them until cold. When I go to peel them I put some warm water in the pan and sit down with the pan and a teaspoon. I take the teaspoon and using the back tap them all over starting at the large end then I sort of roll them between my palms and then pinch up the loose part of the large end and taking the teaspoon run it around the egg between the egg and membrane. The shell comes off almost perfectly. I recently did about 100 eggs this way to make pickled eggs which my DH is eating before the pickle process is done.
Hope this helps