This version of "One Too Many Mornings" is by Jerry Jeff Walker the songwriter of "Mr Bojangles" which was in turn covered by Dylan on his covers album of other artists songs, ironically titled "Dylan"
Dylan wrote this song for Nico who included it on her Chelsea Girl album. Not on any official Dylan albums but is available on some bootleg recordings.
Fantastic thread. Dylan's influence really knows no bounds. Several of my favorite covers already posted but I'll share a couple more. I would also direct your attention to "Chimes of Freedom," a four-CD collection of Dylan covers by a wide variety of artists, from 2012. A handful of the tunes in this thread are from that collection.
Also illustrative of Dylan's reach, a guy put together a series of bootleg recordings called "Nobody Sings Dylan Like Dylan." Each CD featured a string of live recordings of Dylan covers, again by about as many bands as you can name. The series eventually numbered 40-something CDs I think. I've searched quite a bit but either my internet chops suck or the CDs can't be found for download anymore. I've got two of them burned around here somewhere. So good.
The Grateful Dead, and Jerry Garcia in particular, drew heavily on Dylan's catalog. Here's a haunting version of "Senor (Tales of Yankee Power)" by the Jerry Garcia Band--
And for something of a more modern (and obscure) vintage, I'm lately into jamgrass, including the Infamous Stringdusters. These guys can really pick--and Dylan fits perfectly--