In order to continue drawing an existing line, we need to use the line tool and turn on object snapping. This ‘grabs’ the end point of the line and triggers the tool to continue drawing.
Once the line is continued, then we may want to turn off Object Snap. This mode would be turned off (toggled via a hotkey) during the drawing operation.
Unfortunately, I am finding that toggling Object Snap during the operation is cancelling the operation.