Hello all,
It worked perfectly.
Here's what I did, for reference:
1 - Backed-up config.php and MySQL databases
2 - Modified config.php: Made sure $CFG->wwwroot is pointing to the right URL, that is "https://hostname/moodle"
3 - Used the replace tool (http://hostname/moodle/admin/tool/replace/index.php) to find and replace all instances of "http://hostname/moodle" and replace them with "https://hostname/moodle"
4 - Used the purge cache tool (https://hostname/moodle/admin/purgecaches.php) to remove all old references that could still be in there, and force user's browsers to reload content from the site.
Replace "hostname/moodle" in these steps to reflect where your Moodle installation is located.
Big thanks to all of you for your help and time.