All a [self.navigationController pushViewController:controller animated:YES];
does is animate a transition, and add it to the navigation controller stack, and some other cool navigation bar animation stuffs. If you don't care about the bar animation, then this code should work. The bar does appear on the new controller, and you get an interactive pop gesture!
//Make Controller
DetailViewController *controller = [[DetailViewController alloc] initWithNibName:nil
bundle:[NSBundle mainBundle]];
//Customize presentation
controller.modalTransitionStyle = UIModalTransitionStyleCoverVertical;
controller.modalPresentationStyle = UIModalPresentationCurrentContext;
//Present controller
[self presentViewController:controller
animated:YES
completion:nil];
//Add to navigation Controller
[self navigationController].viewControllers = [[self navigationController].viewControllers arrayByAddingObject:controller];
//You can't just [[self navigationController].viewControllers addObject:controller] because viewControllers are for some reason not a mutable array.
Edit: Sorry, presentViewController will fill the full screen. You will need to make a custom transition, with CGAffineTransform.translation or something, animate the controller with the transition, then add it to the navigationController's viewControllers.