By deep I mean with the most obscure original. I am not talking 'all along the watch tower' but things like Fever Ray's cover of Vashti Bunyan's song Here before
Fever Ray is relatively well known while Bunyan had very limited success
“Torn,” by Natalie Imbruglia (1997) was originally written by Ednaswap and recorded in 1995—which, imo, is better. However, even though they wrote it, it was first recorded in 1993 by danish band, Lis Sørensen, who titled it “Brændt,” which means “Burned.”
Bonus! "The State I'm In" a song from Ednaswap’s first album, was covered by Sinéad O'Connor for her 2000 album Faith and Courage.
Harry Belafonte's exuberent cover of "Day O (Banana Boat Song)" which was a hit in its time and re-popularized by the dinner scene in Beetlejuice.
The original is Jamaican folk/traditional and the first recording of it Eric Connor - Day O has a totally different tone to it. In the original, labouring on a banana plantation is as tiring and depressing as you'd think.
Not sure if this applies, but people still seem to think that Nine Inch Nails covered Hurt by Johnny Cash and not the other way round. That or they haven't even heard the original.
Well apparently people think Darius Rucker wrote the Old Crow Medicine Show song Rock Me Mama, and what's more, OCMS in their turn credited it (in jest) to Bob Dylan.
Drives me crazy that people don't know Stomp and Holler is a Hayes Carll song, and that Hallelujah is a Leonard Cohen song, and Downtown Train a Tom Waits song, too.
But I don't know if any of these are as obscure as you are looking for. You are looking for songs even music people don't think of as covers?
John Coltrane's versions of Afro Blue.
Very different from Mongo Santamaría's original. I like both approaches, but Coltrane's recordings of it were all just so powerful.
The Louis Prima Orchestra's cover of "Whatta Ya Gonna Do?", by Sunny Skylar. One of my favorite songs, a perfect blend of New Orleans and New York in the 40s. The link above is the only version I can find on the internet, which is instrumental. The version on the record I have has vocals and is cleaner. I hope somebody works to preserve Louis Prima's catalog from the 40s digitally, there's so much old music at risk of being lost!
There's a kid I know from a few karaoke nights whose go-to song is Beggin. Now it makes more sense why he does that song a bit differently (the first "Beggin" in the chorus he doesn't sing.)
I only grew up with the Madcon version and I didn't know until today that it was a cover.
I also heard a local singer do a really good acoustic cover of that song, where he actually does the rap parts.
SR-71 ended up being a one hit wonder band and the singer ended up being a songwriter and producer, so I could see the song getting shopped around for a second chance.
"House of the Rising Sun" has been covered numerous times. The most known version is from The Animals. I was today years old when I learned that this is the original version of the song. However, my personal favourite cover version is from BTO.
Another well-executed cover version is of the song "Johnny B. Goode", covered by Peter Tosh, beause it is tranferred into a different music genre.
The Breeders "Drivin' on 9" was a cover of song by an obscure band called Ed's Redeeming Qualities. The Breeders version is IMO sweet and meloncholy but the original is rather weird.