As already mentioned, SmartFTP has a feature known as 'server to server' which basically means being able to transfer files from one server directly to another server.

Fire up SmartFTP and log into both servers, that is, the source (where your getting your files from) server and your destination (where the files are going) server. I'm not detailing every click required to achieve this.

With the program there are a number of ways to achieve the one process, as is the case with any modern software. I'll detail the menu way.

Open the files listed in the source server panel and drag and drop the relevant folder into the lower 'temporary queue' panel.

Right-click on the folder name in the 'Temporary Queue' in the bottom panel and select 'Destination' then 'Remote' then your destination folder.

At this point your files should start transferring from the old server to your new server, completely bypassing your local computer.

That's all there is to it, your files will be transferred.

Another source of information is the SmartFTP site itself. Here is a video tutorial relating to server to server transfer of files.