Big Slide Menu - Mobile Push Menu
The first, more tested option is to utilize the Big Slide Menu plugin, which creates a mobile push-menu, which can account for several layers of depth, without demanding too much screen real-estate.
This option may be difficult to integrate the with Drupal Menu system, but it would certainly give you a robust solution, which can accomodate you needs for a large more complex menu system.
Snap.js - Mobile Shelf Menu
The second, but perhaps more complicated option is to integrate Snap.js, which provides a mobile-shelf "type" menu.
To my knowledge it's not a module for Drupal, but that being said you don't have to do much to combine the plugin with drupal.
First, you have to be able to edit the page.tpl.php, which is where you want to add the snap-content selector to page container.
Second, you have to add the Snap content container, which will contain the responsive-ready menu system. I recommend adding this below the page container. You will either want to add a new region in this location ( so you can add blocks from admin/structure/blocks) or create a function to add the appropriate menu to the location. I definitely don't recommend hard-coding the menu in, but it may be a consideration in your case if the menu is static and you want to make the design process a little easier.