Safari's client certificates and related preferences are stored in Keychain Manager with a kind of certificate.
When you select a certificate to use with a web site, it stores another entry in the Keychain Manager with a kind of identity preference. Unfortunately, by default it stores it only for the exact page you were on. Both the name and location are set to the URL of the page.
To fix this, you can just edit one of the identity preference entries and change the where section to the base URL, like
https://somesslsite.com/ (the trailing slash is important!). I also update the name to the same thing to prevent confusion. You can then delete all of the other identity preference entries for that site.
If you have a certificate that expired and you've had to add a new one, I would recommend that you delete the old certificate entries and all of the related identity preference entries.
To find certificate and identity preference entries, open Keychain Manager, make sure All Items is selected, and search for the partial URL and/or certificate name as appropriate. You probably don't have many, so if that doesn't work just sort the list by kind and you should be able to find them easily.
NOTE: I'm answer this myself since I figured it out but wanted to persist the knowledge for myself and others.