Server to Server FTP / FXP

I'm trying to move my websites between different hosts, but the files are enormous and will take literally weeks to upload again. So I'm trying to find a way to do server to server transfers. I know about FXP, but the hosts do not support these.

Is there any way or workaround to do this with FZ or any other software? If it's not freeware, and please PM details so you don't break any forum rules. Thanks