Before trying/doing one of the below, backup the database (sqldump) .... just in case.
First, make sure the site is accessible via port 443 ... https://site/ - yes theme will look wonky but that's a good sign ... web service is reponding to https://
http://yoursite/admin/tool/replace/ .... it's hidden ... not in admin menu.
search for: http://site/ replace with https://site/
Be sure to include the http(s):// in front of 'site' and the closing '/' at the end.
That will warn you ... it's experimental ...I have used it, however, and it does work ... from what I re-call as 2.7 was many moons ago.
The other way
put site in maintenace mode.
Do an sql dump
edit the sqldump with a text editor that has search replace ... same search for replace parameters as above.
Import the edited sqldump into a new DB for Moodle (phpmyadmin or cli). Change config.php to point to the newly imported/edited DB.
With either of above ... change the config.php file URL from http://site to https://site
Clear server caches (purge) clear browser caches and cookies.
Take site out of maintenance mode.
Editing config.php does NOT require restart of web services. Whatever change has been made to config.php if picked up by browser on the very next click.
Then give it a go
'spirit of sharing', Ken