What if you read the directory to get the actual filename? After parsing a file you could then move it to a different directory or delete it so that the next day the file that arrived would be the only one. I'm envisioning something like this:

opendir DIR, "/path/to/mlb/incomingdir" || die "Open dir\n";
@filelist=readdir(DIR);
closedir DIR;
## @filelist has all the files and directories in the given
## directory. You'll want to screen it to make sure you're
## opening a file.
foreach (@filelist) {
  if (-f) {
     &parse_scorefile($_);
     &move_file_to_archive($_);
  }
}
[download]

You can actually compact this, doing the file check at the same time as the readdir, but I separated for clarity's sake. You'll of course want to strictify this.