I see all of the other answers here suggest that you add a A record or a CNAME record to the DNS zone database. In my experience, however, I suggest doing something slightly different.
Instead of just creating a DNS record, there is another way. As it's not a domain controller, it will not have this command by default as it is include with the AD DS role. However, you can install the Remote Server Administration Tools for AD and this command should be included with the installation. After that installation, you can use the following command inside a regular cmd prompt (no need for PowerShell) to not change, but rather add a computer name to the system.
"netdom computername (current FQDN of system) /add (alternate FQDN)"
Note that the above command also works if you are changing the name of the system. In that case, you can perform this command immediately after you perform the previous one:
"netdom computername (current FQDN of system) /makeprimary (alternate FQDN)"
From there, you will need to restart the server for the name change to take effect and for the server to register its changes with DNS.
I suggest this method for two reasons. Firstly, though it may not be the simplest solution, it does ensure that as much as possible will continue to work with the new name. Sometimes, SMB, for example, doesn't play well with CNAME or additional A records. Additionally, even so, a new computer name will register with DNS alongside the regular name of the system and will be dynamically updated with the correct IP address, unlike a manually created A record (though a CNAME record will update correctly).
Additionally, there's another reason - if LLMNR (link-local multicast name resolution) hasn't been disabled on the network and your applications use it to contact the server, using this method will ensure that LLMNR will continue to work with both names because the system will be configured to listen for both names.
EDIT - I read the question wrong. I just realized that the server in question isn't a domain controller and as such doesn't have the 'netdom' command by default as a result of the AD DS role. CNAME and A will work perfectly fine, but in my experience, adding the alternate name to the server has always been better.
EDIT 2 - For this to work, you're going to need to build a new zone, preferably AD-integrated