...
    # this assumes all files in the same dir
    opendir HANDLE, "the_dir"
       or die "couldnt opendir the_dir for reading $!";

   my @files = readdir HANDLE;
   closedir(HANDLE);

   my $most_recent = (sort my_filename_sorter @files)[-1]
   ...

sub my_filename_sorter {

   (my $a_date = $a) =~ s/\.txt$//;
   (my $b_date = $b) =~ s/\.txt$//;

   return $a_date <=> $b_date;
}