Short answer
Very fresh eggs are often harder to peel because the white adheres more strongly to the inner membrane. Age, cooking method and thorough cooling all influence the result.
What changes as an egg ages
Carbon dioxide gradually leaves through the shell and the white becomes less acidic. That change reduces the attraction between the cooked albumen and shell membrane. It is why an older but properly stored egg may peel more cleanly than a newly laid one.
A practical method
Lower eggs carefully into simmering water or steam them, then cool promptly in cold water. Crack the shell all over and begin at the broad end, where the air cell creates a useful gap. Peeling under water can help separate membrane from white.
What an ice bath can and cannot do
Rapid cooling stops further cooking and may help the contents contract slightly, but it cannot completely overcome the chemistry of an extremely fresh egg. If appearance matters, choose eggs that have been refrigerated for several days.