Conversation so far:
What did he do with the Rolling Stones?
Taylor's onstage debut as a Rolling Stone, at the age of 20.
What year was that?
1969.
How long did he play with the Stones?
CANNOTANSWER.
What else happened during this time?
June 1969, John Mayall recommended Taylor to Mick Jagger.
What did Jagger say?
An impressed Jagger and Keith Richards invited Taylor back the following day.

Question: What happened that day?
Self-contained rewrite: What happened the day that Mick Taylor was invited by Mick Jagger and Keith Richards?