Instead of putting a hamburger style menu button on the top corners, could it work well always on the bottom center? It would be visible on almost every screen and free up the top corners for other icons/buttons.
I would not recommend it on Android - this area is very close to the system soft-buttons, so mishitting may happen.
On iOS - even though this is not a standard system pattern - this is some way to go, as the home button is physical, so it needs to be pressed, not just touched. Thus, user error prevention heuristic is not compromised here. There are applications with custom patterns implemented, however, the ones I have seen are only related to new item creation - like this one in Catch Notes app, screens below, with bottom menu contracted and expanded:
(More screens you will find e.g. here - alas, development of this application has been discontinued, and it was a great notes organizer, with color codes etc.).
By the way, Catch Notes was available for Android as well, here's a picture - mind how close the new-item menu is to the system soft buttons: