You know. On top of being dead and then revived, Cesare does have power to him that never belonged to Caligari or to the clowns.
Like, hes been around a VERY long time and has certain perks regarding that. This is the obvious part a lot of people already bring up, but let's start here.
The most obvious Canon one being an absolute fuck ton of money, obviously, with at least one confirmed source (money hole lmao) that is of unspecified origin (fan theory tends to lean toward stolen from db cooper, from some other mysterious source Cesare does or doesn't know of and just found, or that he himself put it there as a stockpile some few ages ago) but is generally known NOT to be from Caligari (I do not remember if that is Worthy confirmed or not, or just hinted, but it IS widely accepted, far as I ever see)
Other obvious perks outside of money include just general experience. He's been around for centuries, it's implied he's been to a lot of places in various hunts for cryptids around the world. He has an eclectic mash of knowledge and skills and experiences.
It's not as Canon Confirmed but it could also be assumed he had some connections that extend from the job. He did, after all, have to hunt creatures that were in hiding.
The point I'm leading up to is that Cesare tackled the hurdle of Steve- immortal and magic- and won. And while some of Cesare’s abilities are directly linked to Caligari, such as the presumed immortality of being a zombie as well as presumed strength and the presumption that the hammer has to be related to Caligari due to it sending steve TO caligari, these aren't the things that, in the end, allow Cesare to succeed. Like, at all, even if he uses the hammer to send Steve down to Caligari once CAUGHT.
The thing Cesare uses to catch Steve is magic kettlebells. The Hard Canon here is that Cesare BOUGHT these kettlebells, expressing how expensive they were in his success.
This can be taken several ways, but the two obvious ones are: that Cesare knows enough about magic to use, as well as make, craft, aquire, steal, etc these kettlebells, or even if he DOESNT know Jack Shit about magic perosnally, then he still has, in some manner, contacts that can and will aquire him magic tools he is capable of employing with, at worst, zero skills in said magic.
Either way, he either knows about magic and how to use it, or at least has access TO magic.
When you fast forward to a post s3 Cesare, he IS going to be significantly more fragile now. He's not undead and he bleeds now and God knows he probably isn't healthy and well and is probably old- but all speculation aside on his physical condition and how much it does or even can improve post revival... removing all theory from the equation.
CANONICALLY. Hard canon? Cesare still has fuck loads of money (potentially more than just Money Hole, he does not stike me as willing to be dependent on One thing, even just in source of money, but again, thats just theory) and either knowledge of magic or magic connections. This is, like, litterally written into canon. Neither of those things were involved with Caligari. Cesare has money the acquisition of usable magic.
So what I'm saying is old man Cesare 100% can still be fucking insane and just have magic bullshit. He could throw another magic kettlebell at anyone he fucking wants. God knows what else he could do.