The best solution would be to call or e-mail the provider of the old system and ask to tar/zip the whole directory tree for you so you only have to download one file, which is easier, much faster and efficient then downloading hundreds of small ones.
But if this is not possible the second best solution would be to login into ssh again on the new server, and install or use an ftp client from within the ssh session that does recursive directory downloading so you don't have to "prompt and mget *" all directories separately.
You can use "sudo apt-get install ncftp" if you have root access to install the ncftp client or upload ncftp3 client sourcecode and compile it yourself with a ./configure and make command
Alternatively you can upload and install the ftp clients yafc, tnftp or quftp to accomplish the same recursive downloading task (standard ftp does not support this).
Another route could be to recursively get the website via http instead of ftp assuming the website is still online or reachable with "wget -r -l0 http://oldsite/" through ssh, but there's a risk you will not get everything.
The last resort alternative is to download everything to your client, zip it and sftp it to the new server to unpack it.