JQuery to the rescue!  The previous web guy had used Superfish, which is a fine system on its own, but he’d only used the bits and pieces he really needed.  Thus, when two-level nested menus became three, the script was not able to handle it.

First step: Use what they already had without reinventing the wheel:

<?php wp_list_pages('title_li=&depth=2'); ?>

Became:

<?php wp_list_pages('title_li=&depth=3'); ?>

And Christian at Kresi.at provided a nice starting point for me to jump from using JQuery in WordPress menus.  He really should be applauded for such a simple solution.  I just had to style the 3rd tier menus and I was set!  (Text blurred to protect client identity).

Nice.

I also managed to wrangle in Image Uploading issues – caused by two separate problems on two different sites.  Freeing up a bit of file permissions did the trick on one, and discovering that there was an entire duplicate WordPress installation elsewhere and re-directing the uploads folder fixed the second.

The whole line of fixes and menu additions took a little over 3 hours.  A total success for myself and the client!