Is there a way to transfer large files from one host to another host? without having to download it to my local laptop and then upload to new host?
Any FTP utility? Or SSH shell command?
If both servers support the FXP protocol, you can use an FXP client to transfer files from one server to the other.
Is this a Linux server? Are you familiar with the Linux shell system?
Most Linux distros will ship with the “ftp” client, just enter the command ftp <server> on the shell prompt of the server you want the files to be transferred to. Then log into the ftp account on the server you want the files transferred from. Use the get command to download files. Type help for more information about the command-line ftp program.
You can do this with ‘backup’ option available in cpanel.
Good luck.
Is there a way to transfer large files from one host to another host? without having to download it to my local laptop and then upload to new host?
Alternatively, you could ask the host to do it for you, though they might refuse for this not being covered by their customer support service. Also, depending on the situation, you might not want them to know that you’re putting these big files on their servers, so that they don’t flag you as a “server abuser”. 
If you have a shell on one server, you could cheat and ‘wget’ the file from one of the other, if not either SCP (Secure Copy) or as has already been said, FXP.
As Tim said, create the large file on one server and wget it from the other. Fairly simple!
Curl is your friend!
Always back up while you transfer on cpanel
if both have cpanel installed, then its easy.
if what you mean here is moving all your site to new web hosting provider, then it will be easy if you find new web hosting that provide free migration service.
they will do it for you with no charge.
The backup option on cPanel is your best option for sure if the transfer is from cPanel to cPanel. The reason for this is because it backs up and restores your databases as well.
Otherwise, a straight forward drag and drop FTP restoration is your second best bet.
If the host does a full account copy, the cPanel system will pull across everything - email setup, home directory, databases, permissions, the lot. This is by far the easiest option but it has to be done by the host (logged into WHM as root). Many hosts will do this for free or minimal charge for a new customer.
The other option is to package up your files on one end and then use wget or similar to pull them across.
eg: zip them up on one end, then on the other end call a PHP page containing something like this:
<?php
...
system ("wget someotherhost.com/yourfiles.zip");
system("unzip -r yourfiles.zip");
...
?>
Some hosts disallow system(), if so, you need to get their help with the copy or find another host.
As others have said, some hosts may support fxp, although when i last tried it I had no success at all. If you do get FXP to work, please let us know.
yes cpanel backup and move over, pretty easy to do and can get you moved from host to host in minutes.
You can do it manually, it’s just your internet which has to be fast.