You don't always want the hint because if you're jumping d steps ahead, you actually need to touch 2 log(d) nodes, once to walk up, and another to walk down. This means anytime you're moving more than d=sqrt(n) steps away you're better off just walking down from the root instead.
For example if n=1000000, if the hot section is wider than d=1000, you're better off going from root all the time. Of course if you access stuff that are directly adjacent you should still use inorder iteration for those (but I see that as a ++ operator, not a separate lookup).
https://en.cppreference.com/w/cpp/container/map/insert
In common implementations iterators are just node pointers, which is enough because nodes have parent pointers, so you can recover the whole path.
In extreme cases like ordered insertion this effectively shaves off a log(n) factor.