OK, I had been looking at the information for “The Loop”, but what I was hung up on was that fact that it was working on one page, but not on the rest. This inconsistency didn’t make any sense to me.
So in “footer.php” I added…
while (have_posts()) : the_post();
… before the line reporting the last updated date, and then…
endwhile;
… after, and the updated dates now show up on all pages rather than just the home page.
Thanks for pointing me in the right direction.