If you process the files in alphabetical order, the time is not so very useful to determine which files aren't processed yet. You have to sort the files by mtime and store the newest time somewhere.

Another possibility: On DOS/Windows, there is an "Archive" file attribute, which is set whenever a file is modified. If your backup software doesn't make use of it (ask your sysadmin), you could clear this attribute for files you processed and filter them out. See Win32::File.