My Grandma used to make deviled eggs that we loved. She would boil them, put them in cold water till they were cool enough to handle. Then she'd gently crack them all over, and pinch off a tiny bit of shell. Then she'd slip a small "tea" spoon the under the skin and slide it all around the egg, The whole skin would slide off. I still do that today with no problems.
As a child I never noticed if she kept her older eggs for this, but today I'm sure that our "boughten" eggs are less than 3 months old. A search on the Internet will tell you that. I let some eggs set out over night to become room temp, because I forget to do this before peeling them.