my @entries;

...

# Loop over your files and extract data

  push @files, {
    'array_position' => $array_position,
    'run'            => $run,
    'copy'           => $copy,
    'district'       => $district,
    'total'          => $total
  };

...

my @sorted_files = sort {
  $a->{'run'}      <=> $b->{'run'}      || # use '<=>' for numbers
  $a->{'copy'}     <=> $b->{'copy'}     ||
  $a->{'district'} <=> $b->{'district'} ||
  $a->{'total'}    <=> $b->{'total'}
} @files;