If your theme includes a header hook, you can simply hook in a new widget in that area and include the conditional tag for logged in users which only displays the custom menu widget with the links you want.
<?php
if ( is_user_logged_in() ) {
wp_nav_menu( array( 'theme_location' => 'logged-in-menu' ) );
} else {
wp_nav_menu( array( 'theme_location' => 'logged-out-menu' ) );
}
?>
You could use this code above directly in your header.php file or remove the PHP tags and include it in a custom function with your theme specific hooks (If has one) and use it in your child themes functions file.
Another option is to create a new widget area and use the widget visibility or another conditional widget which enables you to add the conditional directly to the widget.
If you can provide the theme specific header hook, i can provide the code to create a new widget area.
Source http://codex.wordpress.org/Function_Reference/wp_nav_menu
function wpsites_register_header_widget() {
register_sidebar( array(
'name' => 'Header Widget',
'id' => 'header-widget',
'before_widget' => '<div>',
'after_widget' => '</div>',
) );
}
add_action( 'widgets_init', 'wpsites_register_header_widget' );
add_filter( 'your_themes_header_hook', 'wpsites_header_widget' );
function wpsites_header_widget() {
if ( is_user_logged_in() && is_active_sidebar( 'header-widget' ) ) {
dynamic_sidebar('header-widget', array(
'before' => '<div class="header-widget">',
'after' => '</div>',
) );
}
}
Drag in a custom Menu Widget and you’re set.