It is very common and RECOMMENDED to leave your OLD QC server still available (but de-activate all projects) during an upgrade.
HPE supports having the 2 servers active during the upgrade - AND you can indicate during install to use an ''evaluation license'' for the NEW server until you obtain a newer ''migrated'' license for the newer version of QC (you will need log a case with HP-support to get a ''migrated'' license for a specific QC version).
As a side note, when you start upgrading your projects, it will be comforting to know that if you have a project or two that have issues upgrading, you can fall back on the old server (just activate projects as needed) until you resolve the individual project-level upgrade issue.
When you figure out what to do for the trouble project upgrade, you can de-activate the project in the old server, re-migrate that single project and repository again to the NEW environment and upgrade it.