In interest of full disclosure, opinions are mixed about how well TOCE meshes with the original set. A lot of people get hung up on the fact that Cthulhu factions can pass out madness, and original factions aren't equipped to get rid of them.
But I agree with J. There's actually only two TOCE factions that pass out madness (Cultists and Elder Things) and only ONE that has cards that get rid of Madness (Miskatonic U). Like all of Smash Up, a lot depends on the faction you choose to counter your opponents'. For instance, since Wizards can bank up extra plays, they're actually a great counter to madness, which can be trashed just by playing them. And Wizards are a base faction!
The fact that you can spend your action for a turn to return a madness card to the deck makes it so all factions can get rid of madness cards.