If your hosts are both cpanel you can try this steps
This is the most easiest method used to transfer whole site backup from server to server. It makes use of the "Backups" feature in cPanel and needs to be enabled by your host.
To use this
Open the Backups manager in your cPanel and click "Generate/Download a Full Backup"
For Backup destination choose "Remote FTP Server (Passive Mode Transfer)"
Enter the FTP server, username and password
Enter the FTP server port (this is usually "21")
Leave Remote Dir field empty
Hit "Generate Backup"
Your backup will now be passively generated and transfer from server to server using the net2ftp protocol. If you don't see this option in your cPanel then you need to ask your host to enable it.