When you need to move the SQL Server instance to another server, you can use SQL Alias to get avoid of changing connection string for softwares which use this database.
All you need is setting up SQL Alias on the computers running your softwares. Nothing need to be done with the SQL Server if it’s installed other than the computers running applications.
The tool for setting up SQL Alias is included with Windows. You just need to run cliconfg.exe to set. Be careful the typing — it’s not config.
If your Windows is x64 based, you may need to choose the right version of that tool. The x64 version is C:\Windows\system32\cliconfg.exe and the x86 version can be located as C:\Windows\syswow64\cliconfg.exe. If you don’t know which is the right one, set up both of them.