my @folders = glob 'input_directory*'; # get/sort all qualifying folders
my @requests;
for (my ($found, %filedate); $found = pop @folders or exit();) {
    @requests = sort {
      { ( $filedate{ $a } ||= -M $a )
      <=> ( $filedate{ $b } ||= -M $b )
      } glob "$found/" . 'Flat_file.*.txt'
        and last; # reiterate iff none found
}