There are two main mistakes that authors make when it comes to raising the stakes from one story to the next.
The first is that they either raise the stakes too quickly, or simply don't know when to stop raising them, and the result is that eventually they run out of room in which to escalate without things just getting ridiculous. And as you mentioned in the question, once you get to that point, it's very hard to dial things back again without your audience getting bored.
The classic example of this would be the Dragon Ball franchise. It started off as just the goofy adventures of a kid with a monkey tail. Then it started introducing antagonists that threatened to take over the world. Then Piccolo actually did take over the world. Then villains started showing up that were capable of destroying worlds, and again, they actually did so. Then they started facing off against literal gods that were strong enough to destroy the entire universe, and now (IIRC) they're fighting beings powerful enough to destroy the entire multiverse. Where do you escalate to from there?!
The second problem is that, more often than not, these newer, bigger threats tend to come out of absolutely nowhere, with no build-up or foreshadowing. Sometimes, this can work well and surprise both the audience and the characters, but other times, it makes it look like you're just pulling these new threats out of your arse (which, a lot of the time, is actually the case). I recall that one Tumblr post about Supernatural:
It's like, you beat the Devil himself? Well, now you've gotta fight the Devil's cousin Phil, who has conveniently gone entirely unmentioned up until now, but he's totally twice as evil.
That last paragraph was literally supposed to be the most ridiculous hypothetical example I could think of, and people are messaging me to say "His name was Metatron, not Phil". I can't even make fun of this show.
So how do you avoid those problems? Well, the first one can be avoided by making sure the stakes are raised gradually, and by mixing in more personal threats. For example, Book 3 might have the hero fighting to stop a villain from conquering the world, but Book 4 might have him fighting to save a loved one from a curse/illness/kidnapping/[insert crisis here]. For the world as a whole, the stakes are much smaller in Book 4, but for the hero, the stakes are higher because the threat is that much more personal.
The second is easy to avoid if you're planning all six books in advance, and can build up each threat before they appear, but what if your publisher suddenly asks you to write a seventh book? Then an eighth, and a ninth? You have to keep making up new threats as you go, and that's where this problem tends to creep in. You just have to try and make sure that each new threat is consistent with the world you've built up to that point, and that it makes sense that none of your characters have heard of it or mentioned it before. Dragon Ball, for example, is able to sidestep this problem by having its villains come from outer space, or alternate universes, or the future - of course Goku hasn't heard of the Androids, they don't exist yet!