Your custom container view controller will need to adjust the contentInset of the second view controller according to your known navigation bar height, respecting the automaticallyAdjustsScrollViewInsets property of the child view controller. (You may also be interested in the topLayoutGuide property of your container – make sure it returns the right value during and after the view switch.)
UIKit is remarkably inconsistent (and buggy) in how it applies this logic; sometimes you’ll see it perform this adjustment automatically for you by reaching multiple view controllers down in the hierarchy, but often after a custom container switch you’ll need to do the work yourself.