Sysadmin How to extend hard-drive space?

If you have to ask, then it's not task you should be running. Safest method will be upgrade to a larger server with larger disk capacity and migrate data over.

For moving site data there's 2 general guides where the principles & steps of backing up and moving data is the same just slightly different for. You'll want to be using 1st guide.

1. Moving Centmin Mod site data to another Centmin Mod server
2. Moving cPanel/WHM site data to Centmin Mod server

Those guides move alot more than just the site(s) in question but config files etc. You can use your own best judgement as to the config files and whether you want to move them or just use the fresh server setup Centmin Mod config files as they get auto optimised for the server Centmin Mod initially installed on.