Tom Riddle finishes his seventh year and applies for a job teaching at Hogwarts

Headmaster Armando Dippet turns him down, saying that at eighteen he is too young. He then goes to work for Borgin and Burkes instead.

Timeline Notes

We don't know when this happened exactly, except that it was almost certainly within a few weeks of the end of school. Riddle gets the job at Borgin and Burkes "the next thing the staff knew," according to Dumbledore.