Yup, it's been possible since the very first version of Admin Tools in 2010.
Requirement: both domain names (www.old-website.com and www.new-website.com) must be served by the same hosting user account. For cPanel–based hosts this means that one of these domains is the primary domain of the account and the other one is an add-on domain.
In this case you can go to the .htaccess Maker and find the “Redirect this (old) domain name to the new one” option. Enter your old domain name in there. Example:
Every time someone tries to visit the old domain name the request is handled by your server. However, the .htaccess file redirects it to the same URL on the new domain name.
You can try this on our own site. Between 20010 and 2020 our site was www.akeebabackup.com. Ever since September 2020 it's www.akeeba.com. Try visiting the URL of your ticket on the old domain i.e. https://www.akeebabackup.com/support/admin-tools/37513-rewrite-base-url.html. See what happens? You are magically redirected to the same URL on the new domain name. That's how we changed our domain name and nobody, literally nobody, noticed or did even as much as bat an eyelid.
We applied the same trick in 2010 when we moved from our old site (joomlapack.net) to akeebabackup.com. Again, most people never noticed the change. Those who did notice the change only did because there was a very big difference in the domain name, a new site design and we started selling subscriptions. Most people really only noticed the latter :D
Nicholas K. Dionysopoulos
Lead Developer and Director
🇬🇷Greek: native 🇬🇧English: excellent 🇫🇷French: basic • 🕐 My time zone is Europe / Athens
Please keep in mind my timezone and cultural differences when reading my replies. Thank you!