How can I transfer large files from
new-domain.com(new-host)? Is it possible to transfer it without downloading the files from my old-host to my local pc? I have an FTP details of both.
From my experience, wget is the ideal tool for your task. You can tweak it hundreds of ways to download exactly what you need in just the way you need it. You can even download to your new server the entire website with one single command. Wget also has features to efficiently try again if it was unsuccessful or if the file was only partially completely downloaded.
I use wget every time my Internet connection is flaky and I need to download a backup of my website. It has failsafes to get the job done in harsh conditions.
Or, for "special" big files that aren't accessible via a regular URL, you could establish an actual FTP connection in the shell of your new account and have the server directly pull down the big files that way.
SCP is ideal for this, SCP Usage Documentation: