If 'LIST -alR' is unable to show files/folders from sub-directory's it means that the ftpd (ftp server) does not support the -R (recursive) extention. This means that there is no quick way to do what you are asking with flashfxp or any other ftp client that does not have good scripting support (using a custom script to go through all of the folders would take just as long as doing it manually).
If the server uses dated folders to organise new content, then i assume that the content of folders for previous days will not change so you would not need to check them very often.
If your purpose of going through many folders is to search for something specific then you can simply use 'Tools > Find Files on FTP Server'
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