Bootstrap 4
Create a responsive navbar sidebar "drawer" in Bootstrap 4?
Bootstrap horizontal menu collapse to sidemenu
Bootstrap 3
I think what you're looking for is generally known as an "off-canvas" layout. Here is the standard off-canvas example from the official Bootstrap docs: http://getbootstrap.com/examples/offcanvas/
The "official" example uses a right-side sidebar the toggle off and on separately from the top navbar menu. I also found these off-canvas variations that slide in from the left and may be closer to what you're looking for..
http://www.bootstrapzero.com/bootstrap-template/off-canvas-sidebar http://www.bootstrapzero.com/bootstrap-template/facebook