Moving to a new host I need to copy big files and my upload speed is slow to do that from my PC. How can I copy from old ftp server (PHP) to new one directly ?
If the two FTP servers support the File eXchange Protocol, you may do this using an FTP client that supports it.
An example is FTP Rush, a free program for Windows that is available as a portable version and setup version.
For a review, see the article Transfer files directly between servers with FTP Rush.
If one of the servers does not support this protocol, you could possibly use a cloud service such as MultCloud, where the free account lets you copy up to 50 GB of data.