Back in the 50's and 60's, it was much more common for people to put out their own version of a hit song. Some of that was due to race. because certain radio stations would not play records by black artists, people like Pat Boone would cut 'white' versions of R&B hits.
And there were singers who did not write their own material who would cover songs by lesser-known performers - i.e. Linda Ronstadt covering songs by Warren Zevon and Elvis Costello.
One of the more interesting ones was the Eagles covering "Ol '55" by Tom Waits on the 3rd Eagles album, "On the Border." (Waits did not like it....)
a lot of blues musicians tend to do their own versions of blues classics. like Voodoo Child by Hendrix - great cover by Stevie Ray Vaughn.