The swampy jam was recorded in October 1974, and features Blind Faith’s Ric Grech on bass.
It’s set to appear on the deluxe reissues of the Stones’ Goats Head Soup, which is due for release on September 4.
Keith Richards says of Scarlet’s recording: “My recollection is we walked in at the end of a Zeppelin session. They were just leaving, and we were booked in next and I believe that Jimmy decided to stay.
“We weren’t actually cutting it as a track - it was basically for a demo, a demonstration, just to get the feel of it but it came out well. With a line-up like that, you know, we better use it.”
Mick Jagger adds: “I remember first jamming this with Jimmy and Keith in Ronnie’s basement studio. It was a great session.”
The Goats Head Soup reissue features three previously unheard tracks - Scarlet, Criss Cross and All the Rage - as well as Brussels Affair, a 15-track live album recorded in Belgium on the Stones’ autumn 1973 tour.
It’s available to preorder now from The Rolling Stones’ website.