Covers are interesting subjects to discuss; while some are untouchable, others can emerge entirely different (and in many instances better). I think there are generally two categories for covers - Straight Covers (with little/no changes; retains a lot of the original spirit/intent), and Cover Versions (which have changes that are noticeable in terms of vocalization, musicianship, etc.). Something like 'Hallelujah' is fine when sung by Leonard Cohen, but better when sung by Rufus Wainwright or Jeff Buckley. Sinead O'Connor's version of 'Nothing Compares 2 U' is so good no one cares that Prince wrote and performed it first.
A song I heard recently, that I thought was amazing because I'd never heard it before, is Ana Gasteyer's cover version of 'Book of Love'. I did some research and found that it was best known as a Peter Gabriel song, but that he did not write or originate it - it was written by Stephen Merritt of The Magnetic Fields and released in 1999. Gabriel covered it in 2004. Gasteyer then covered it for her debut album in 2019. Her version completely changes the mood/tone of the song; Gabriel's is great, but almost feels like spoken word. She harmonizes more and really sings it - just a tremendous effort that may be the best love song I've ever heard (up there with Kate McGarrigle's 'The Walking Song').
Peter Gabriel
Ana Gasteyer