If you need to add SSL hostnames to vCenter or have it work as a different hostname than when you installed vCenter, you can take advantage of a configuration file called webclient.properties. Before we begin. Take a backup of this file.
Stop your vCenter UI service:
service-control --stop vsphere-ui
Then open the file located at /etc/vmware/vsphere-ui/webclient.properties and look for this line. Uncomment it and place your alias after the equals sign. Then save it.
sso.serviceprovider.alias.whitelist=<your fqdn or shor hostname>
After saving the file, start your vsphere-ui service back up:
service-control --start vsphere-ui
Make sure you have your DNS configured to point this new hostname to your vCenter IP address or a proxy that is sitting in front of it that is handling SSL termination.
Troubleshooting
If the service doesn't start, make sure vsphere-ui is the owner of the webclient.properties file. As a side note, I was able to put vCenter Server behind a Traefik proxy for SSL certificates in front of vCenter using this method.