Redirect to first child

Hi,

I’ve found this template code in another topic, here.
It is perfect for what I want to do.
My only problem is that it applies only to 2nd level children.
I would love to have it redirect to 3rd and 4th also.

Parent > Redirects to… (OK, works)
– children 2nd > redirects to… (not OK)
– children 3rd > redirects to… (not OK)
– children 4th

I have no idea why it doesn’t work currently, since it should be like a chain of events (where every children is considered as a parent, and such, if the template is applied, should redirect descending automatically?

Many thanks in advance for your help.

<?php
/*
Template Name: Redirect To First Child
*/
if (have_posts()) {
  while (have_posts()) {
    the_post();
    $pagekids = get_pages("child_of=".$post->ID."&sort_column=menu_order");
    $firstchild = $pagekids[0];
    wp_redirect(get_permalink($firstchild->ID));
  }
}
?>