Thanks for the reply, Till.
If you select * in the website settings, then you can access the site from both IP addresses.
I require a dedicated IP address for SSL on this particular domain. And this box is configured to support SNI, so there are dozens of other websites that are bound to all interfaces ("*"), each with a unique/proper SSL certificate.
Does that mean selecting "*" this is not an option in this particular case? What if I change all of the sites that leverage SNI to be bound to a specific IP address (other than the new one I just added to support SSL on this site).
If not, I will try the other steps you suggested. They make sense.