I can think of a few that surprised me but led me down some interesting rabbit holes-

Mystery Train by Elvis (original by Junior Parker)
Hound Dog by Elvis (Big Mama Thorton)
Long Black Veil by The Band (Lefty Frizell)
If You've Got the Money Honey by Willie Nelson (also Lefty Frizell)
Time is on My Side by the Rolling Stones (Irma Thomas)
It's All Over Now by the Rolling stones (Bobby Womack)
You Were On My Mind by (don't remember) (Ian and Sylvia, written by Sylvia)