Which region? It makes a difference sometimes how the locals view your realm. The solution I would always suggest is both. Yep, how convenient.
But it's true, if you want a newly acquired region (Judging by your avatar I will assume you are asking about Braga) to regain stats faster, you'll need a concerted combo effort. Full estate coverage, knights of that region with police units doing normal police work, courtiers surveying admin, diplomats praising realm, heroes telling tales, lords/judges holding courts. Sounds involved, sure, but in a realm like Sirion, there is bound to be at least one courtier with free time, and one diplomat with nothing better to do than praise the realm. And if a lord asks his knights to perform some police work in their region, they better do it, or risk getting the boot.