There's no wait the first time you toss. I don't know (especially given SE's penchant for grammatical and specificity errors in their update notes) whether the 27 day wait for a second+ toss, starts counting the 27 days from a. the time you tossed the new ring or b. the time you got the new ring.
Easy (if tedious) way to test: get a new ring (no wait). Wait about, I dunno, 10 days or so. Toss it. See if you can get another new ring in 17 days (27 days after getting the ring) or 27 days (27 days after tossing the ring), after that.