my ($t0, $path0, $out);
$t0 = time;
$path0 = path_for_time($t0);
open($out, ">>", $path0);       # note below
while (defined(my $line = readline())) {
  my $t = time;
  my $path1 = path_for_time($t);
  if ($path1 ne $path0) {
    close($out);
    $path0 = $path1;
    open($out, ">>", $path0);
  }
  print $out $line; flush($out); # note below
  ... perform other processing on $line ...
}