Add a custom menu item that supports wildcards

This code can be used after the new post was saved. Otherwise there is no post object at the time.

Please pay attention that the code was not tested.

<?php
// get post ID from the URL query
$post_id = $_GET['post'];

// get post object
$new_item_to_add = get_post($post_id, OBJECT);

// get menu to add the item into (change to the relevant slug, ID, or name)
$existing_menu = wp_get_nav_menu_object('test-nav-menu-slug');
$existing_menu_id = $existing_menu->term_id;

$new_menu_item_data =  array(
    'menu-item-db-id' => $menu_item_db_id, // default 0, creates a new one
    'menu-item-object-id' => $post_id,
    'menu-item-object' => '', //default
    'menu-item-parent-id' => 0, //default
    'menu-item-position' => 0, //default
    'menu-item-type' => $new_item_to_add->post_type,
    'menu-item-title' => $new_item_to_add->post_title,
    'menu-item-url' => get_permalink($post_id),
    'menu-item-description' => '', //default
    'menu-item-attr-title' => '', //default
    'menu-item-target' => '', //default
    'menu-item-classes' => '', //default
    'menu-item-xfn' => '', //default
    'menu-item-status' => 'publish',
);

wp_update_nav_menu_item($existing_menu_id, $menu_item_db_id = 0, $new_menu_item_data);

See wp_update_nav_menu_item code reference.

Also, there is a worse way. You can alter the menu on-the-fly. This will not update the menu:

<?php
function wpse267737_add_nav_menu_items($nav_menu_items) {

    // get post ID from URL
    $post_id = $_GET['post'];

    // get current post data
    $link_url = get_permalink( $post_id );
    $link_anchor =  get_the_title( $post_id );

    // build new menu item
    $new_menu_item = '<li><a href="' . $link_url . '">' . $link_anchor . '</a></li>';

    // concatenate existing items and new
    $nav_menu_items .= $new_menu_item;

    return $nav_menu_items;
}
// change TEST-MENU-SLUG to the relevant menu slug
add_filter( 'wp_nav_menu_TEST-MENU-SLUG_items', 'wpse267737_add_nav_menu_items' );

See wp_nav_menu_items reference.