If all you want to do is increase the number of licensed users, and change the address of the site, you technically do not need to install Jira afresh. But you can do that too.
Once you have the new license key from your https://my.atlassian.com site, you can take that key, copy it, and then follow the instructions in Licensing your Jira applications in order to apply this new license key to your existing Jira instance.
Hosting the site under a different address is typically something that your network admin would help with. Most public facing Jira instances are sitting behind a reverse proxy, load balancer, and/or a firewall. In cases like that, you might not need to actually change settings in jira to make that work, it tends to be settings on the proxy/loadbalancer/firewall that have to be changed in order to handle traffic to the new domain and then redirect it to the Jira instance.
Of course you are welcome to create a new installation of Server instead and then move your data there. In those cases, I would recommend reviewing Backing up your Data first. I would also recommend the guide on Migrating Jira applications to another server. This method is the preferred way to transfer all your Jira data to another instance.
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