The function you are using 'menu_navigation_links' only displays links for a single level. You would be better looking at either the menu block module (https://www.drupal.org/project/menu_block) or using functionality such as the example below:
/**
* Get a menu tree from a given parent.
*
* @param string $path
* The path of the parent item. Defaults to the current path.
* @param int $depth
* The depth from the menu to get. Defaults to 1.
*
* @return array
* A renderable menu tree.
*/
function _example_landing_get_menu($path = NULL, $depth = 1) {
$parent = menu_link_get_preferred($path);
if (!$parent) {
return array();
}
$parameters = array(
'active_trail' => array($parent['plid']),
'only_active_trail' => FALSE,
'min_depth' => $parent['depth'] + $depth,
'max_depth' => $parent['depth'] + $depth,
'conditions' => array('plid' => $parent['mlid']),
);
return menu_build_tree($parent['menu_name'], $parameters);
}
This function will return a menu tree starting at a given url for a given depth.
The following code will produce a sub menu, with the current page as a parent and show child items up to a depth of 2.
$menu = _example_landing_get_menu(NULL, 2);
print render($menu);