In my experience, the learning curve happens in the current role, as a prerequisite to transitioning into the next. You have to be operating at the next level already and there really isn't an easing of the transition beyond that. For example, managing wider scope, solving harder problems, navigating trickier interpersonal dynamics, connecting more "dots", communicating with more clarity on more complex matters, and influencing more people and outcomes, are among some of the skills needed at every PM level, but to different degrees. As you showcase these skills you are already solidifying the foundation for the next level. One of the biggest learning challenges can however come at later levels - when transitioning from IC to manager (irrespective of the accompanying title), partially because it is a binary situation. You are either a manager of people or you are not; there is not much preparing or gradual transition.
Relatedly, many PMs are seeking that check-list, that when done, spells promotion. Consequently, one of the most frustrating and non-actionable things that managers can say is "you are not ready yet", perpetuating a bit of the feeling that there is some transition or steep learning that one must overcome. Often it means the manager is not ready to do that for you but is uncomfortable giving you specifics. Or it is not possible in the current org strucure. Sometimes, they just don't know themselves, but they'll know it when they see it. Perhaps, that is a big hint. Observe how others at the next level are operating as preparation. How big are their projects? How much more do you have to know or do? The criteria can be very different by company, that's why it's important to know any published leveling criteria. But generally, even then, there aren't a lot of clear-cut check boxes. Sometime the criteria are only in your managers head. I once had a manager tell me that if I grew my product to $50M or $100M in revenue it would get me promoted. (My answer was to remind them that we were working on 5-year incubation, hence low likelihood of multimillion dollar outcomes on any reasonable timelines, and that this criteria should have been articulated a lot earlier in the job decription. Plus there were plenty of examples of promotions without that criteria - so the aforementioned criteria disappeared from our conversations.)
In all cases, cross-reference everything official with what you actually see happens day-to-day and what behaviors or results are rewarded at the next level. Perhaps, the steepest learning curve is learning what specifically gets you promoted at your particular company and with your particular management chain.