There is no rule. The Pens fired Therrien less than a year after a Stanley Cup Finals appearance. The Devils fired Julien after a 47-24-8 record just before the playoffs.
The team does what is best for it, especially over the long term.
I'm gravitating towards the conclusion that WS had: if the coach has to speak French, then the best one possible is Guy Boucher, and the window is fairly restrictive to land him. At the very least, I would hire him as an assistant, and impose him on MT. MT must be made to consult with him, and if he does not like it, he can resign. I would try to buy MT out anyway; wonder if there is a buy-out clause in these coach contracts.