I am transferring my WordPress site. I need to transfer all the content to my new server from old one. I don't have that enough bandwidth that I can first FTP to my local PC and than again FTP to the new server. So I request people right here to help me find out some sophisticated technique.
You should ask your new host for help, as they will have enough bandwidth and probably ample experience with this. They might charge extra for this. Moreover, they should have a function that lets them transfer directly from the other host if they both use WHM and don't block this option (most hosts do).
In general, the standard way of transferring anything between two Linux servers is rsync. Here is a simple guide: http://johnveldboom.com/posts/sync-files-between-servers-using-rsync/
However, this may be impossible on shared hosting (as well as any other SSH-based transfer).
The only other way I can think of without downloading the files to your computer is to compress everything on the original host (don't forget to delete after you're finished!), and download it on the destination host. You will have to write a script and put it in the destination server.