As much as it sucks, I'd bet there are quite a few NFL teams that are also in the same boat. To be fair, the model for many teams really hasn't been to hire prior HC.
Many teams hire retreads which wind up sucking (think Shannahan and Gibbs 2.0 in Wash).
There's not magic bullet. Some retreads do well, some don't. Fox is doing well, but he has #18. Carroll is doing well, but he got a team loaded with talent and lucked into a 3rd QB who is playing great. Belicheck is a retread and might be one of the greatest ever. Doesn't hurt that he has arguably the greatest QB to lace them up his entire tenure there.
Caldwell is a retread and can thank Schwartz for that team. He'll be one and done in the playoffs. Reid is a retread and is out (he inherited a team with what, 9 pro-bowlers?).
Fisher is a retread and his entire career is mediocre
Coughlin is a retread and has won 2 SBs, but he's also 'fired' every season. Bizarre.
Lovie is a retread and has the worst team in the NFL.
Wisenhunt is a retread and has the second worst team in the NFL.
There are 12 teams in the playoffs. Only 4 of those teams feature a team with the former HC at the helm - and one of those is Belicheck.
While I do think it's time to change the game, hiring a retread appears to have about the same success as hiring a hot shit coordinator.