In the 6th chronicled year at Hogwarts, Snape is the new Defense Against the Dark Arts (DADA) teacher. Slughorn replaces him as a Potions teacher. At the end of the sixth year
Snape kills Dumbledore, per Dumbledore's request. Nobody except for Snape knows that though.
The next year
Snape replaces Albus Dumbledore as headmaster of Hogwarts.
And Amycus Carrow becomes the new Dark Arts teacher. Now, all the DADA teachers (at least the ones mentioned in the book, while HH&R were at Hogwarts) could not have been DADA teachers the next year.
Memory was wiped nearly completely.
Lupin turned into a werewolf on school grounds. The incident was found out by parents of Hogwarts students, who wouldn't allow Lupin to teach there anymore.
Actually Barty Crouch Jr.
Was dealt the Dementor's kiss.
After everyone figured out that Dumbledore was telling the truth about Voldemort's return, his reputation was reinstated and therefore Umbridge was removed from Hogwarts.
The problem is that Snape could've taught DADA in what would've been HH&R's seventh year, there was nothing stopping him. If somebody else had become
instead, then Snape could've taught DADA two years in a row. Why would Snape not take the DADA job and break Voldemort's curse?
Feel free to edit the spoiler tags. I just wanted to be safe.