Custom adaptation for my sub-menu script that treats first-level items as root-level items. Full documentation here: http://christianvarga.com/how-to-get-submenu-items-from-a-wordpress-menu-based-on-parent-or-sibling/

functions.php

<?php

/** wp_nav_menu( array(
      'theme_location' => 'primary', 
      'sub_menu' => true
    ) );
    
    or
    
    wp_nav_menu( array(
      'theme_location' => 'primary', 
      'sub_menu' => true, 
      'show_parent' => true
    ) );
    
    do not use direct_parent!
    
**/

// filter_hook function to react on sub_menu flag
function my_wp_nav_menu_objects_sub_menu( $sorted_menu_items, $args ) {
  if ( isset( $args->sub_menu ) ) {
    $root_id = 0;
    
    // treat first level items as root level items
    $root_level_items = array( 0 );
    foreach ( $sorted_menu_items as $menu_item ) {
      if ( $menu_item->menu_item_parent == 0 ) $root_level_items[] = $menu_item->ID;
    }
    
    // find the current menu item
    foreach ( $sorted_menu_items as $menu_item ) {
      if ( $menu_item->current ) {
        // set the root id based on whether the current menu item has a parent or not
        $root_id = ( $menu_item->menu_item_parent && ! in_array( $menu_item->menu_item_parent, $root_level_items ) ) ? $menu_item->menu_item_parent : $menu_item->ID;
        break;
      }
    }

    // find the top level parent
    if ( ! isset( $args->direct_parent ) ) {
      $prev_root_id = $root_id;
      while ( ! in_array( $prev_root_id, $root_level_items ) ) {
        foreach ( $sorted_menu_items as $menu_item ) {
          if ( $menu_item->ID == $prev_root_id ) {
            $prev_root_id = $menu_item->menu_item_parent;
            // don't set the root_id to 0 if we've reached the top of the menu
            if ( ! in_array( $prev_root_id, $root_level_items ) ) $root_id = $menu_item->menu_item_parent;
            break;
          } 
        }
      }
    }

    $menu_item_parents = array();
    foreach ( $sorted_menu_items as $key => $item ) {
      // init menu_item_parents
      if ( $item->ID == $root_id ) $menu_item_parents[] = $item->ID;

      if ( in_array( $item->menu_item_parent, $menu_item_parents ) ) {
        // part of sub-tree: keep!
        $menu_item_parents[] = $item->ID;
      } else if ( ! ( isset( $args->show_parent ) && in_array( $item->ID, $menu_item_parents ) ) ) {
        // not part of sub-tree: away with it!
        unset( $sorted_menu_items[$key] );
      }
    }
    
    return $sorted_menu_items;
  } else {
    return $sorted_menu_items;
  }
}