At the bottom, https://gethttpsforfree.com/ links to its source at GitHub - diafygi/gethttpsforfree: Source code for https://gethttpsforfree.com/. There you can see:
TODO (pull requests welcome):
- renew.html - A page with steps for renewing certificates
So it seems like the renewal page is just not done yet.
If you want/can use the letsencrypt-auto program instead, this worked for me (but alas I am not sure if it also works if you didn't use letsencrypt-auto in the same directory when you got your first cert):
To test it:
./letsencrypt-auto certonly --renew --email my.email@example.org -a manual -d my-domain.example.org --dry-run --agree-tos
It will ask you to put a file on your server, that's probably the same as step 4 of gethttpsforfree.com.
To really run it:
./letsencrypt-auto certonly --renew --email my.email@example.org -a manual -d my-domain.example.org
You can run this on a machine that is not your web server. You only need to put the verification file on the web server.