This is about the 1957 Desmond - Mulligan Quartet that recorded tracks Aug 2nd and 27th, released on the self-titled Verve LP (later also known as Blues In Time). Joe Benjamin plays bass, and Dave Bailey, drums. That record and all subsequent versions on vinyl and CD has a track called Standstill (or sometimes Stand Still) that is 3:24 in duration and composed by Gerry Mulligan. I have not found reference to another Standstill in any of the various discographies or other sources.
What I have found on the 1962 Verve Compilation 3-LP set Greatest Names In Jazz, credited to the quartet, is a different Standstill that is 8:16 in duration, composed by Paul Desmond. It does not seem to be Standstill at all, but in fact an alternate take of the song Battle Hymn of the Republican. The song by that title (7:42, Desmond) was also released on the 1957 LP, though you can tell with the first few notes that this compilation track is quite different. It has the same general structure as Battle Hymn, trading sax solos, harmonies, bass solo, etc. The compilation was released both in mono and reprocessed stereo, and includes tracks from a wide range of Verve artists. I don't know if any of the other recordings are obscure or uncommon.
I'm guessing this take has to be from the same sessions, and surely back in 1962 fans would have recognized this as a new, non-LP track. But since that is a little before I was buying records (or even born), I just have our modern sources to go by. I stumbled upon this track only recently, having been previously satisfied that I had identified all available authorized Paul Desmond (without Brubeck) recordings some time ago. In retrospect, the track duration and composer credit do offer clues that it is something different.
And I'm happy to make an audio file available, if anyone wants a listen?
https://www.discogs.com/release/9406618-Various-The-Greatest-Names-In-Jazz