Including a sub-template in a theme
No need to set up hook_theme, Just include like this:

{% include '@theme_name/system/footer.html.twig' with content %}


this will include that template in your main template.  The variables will need to be available from the main template.  

The "with" statement sets up the variables being sent to the sub-template.  See http://twig.sensiolabs.org/doc/tags/include.html