in regard to:
mywebsite/images/smiles/lol
(three directories deep)
then yes any host should be able to provide you with this. If they don't I'd be interested to find out why

Any scripts actually go a lot deeper than this.
As for 'sideloading' your website from one server to another, I don't think logging into the shell and tar'ing your website then using wget will help

so try a site like
http://www.webattack.com and look at their shareware or freeware areas for ftp software. I don't know of one myself that will ftp from server to server, but I do know there are a few that will do this. The descriptions of the software itself should let you know. VoyagerFTP is coming to mind as one that might do it, but I couldn't say for sure.
Greg Moore