So I created a custom menu through the Drupal administration and named it "More Menu." Its machine name, menu-more-menu, was auto-generated by Drupal.
Creating this menu automatically created its corresponding block, and its now available under Blocks administration as More Menu. I then enabled the block and placed it under my desired Region.
I wanted to theme the output of this block. I tried creating a template by following the template suggestion as described here, block--module--delta.tpl.php.
The template file I created was named block--menu--menu-more-menu.tpl.php.
It doesn't seem to work, and I still cannot override it.
Am I missing something?