Amazing how I've never thought about this 'conundrum' before, but it's intriguing. Here are the possible solutions I could come up with:
Two menu buttons
My first thought was to put the sub nav off-canvas to the left. Leaving you with a menu button on the right for a dropdown menu of the main nav and a menu button on the left for the off-canvas sub nav. But that's the whole problem with it, it leaves you with two different menu styles which is confusing at the least. You could solve this with a secondary header and another drop down menu.
My second thought was to merge the two navigations. You can do this when there is a clear hierarchy with the parent link in the main nav and all child links in the sub nav. The structure will look something like this:
This won't work when the links in the subnav have nothing to do with the links in the main nav. This would however be my pick and would even advice revising your navigational structure.
Leaving subnav visible
The last solution I could think of is leaving the subnav visible on the page, but changing it to a horizontal, smaller navigation. This will put more distance between the content and the main header with the main menu so it might not be preferable for you.
Here is some further reading:
Responsive design Pattern for secondary navigation - is there a way not to swamp the user?