Do you mean you want a condition that has to be met ONCE and then is "sticky"? Like:

my $keep_going = 0;
while (my $job_B = readdir DIR_B) {
  print "D\n";

  # ||= here means "assign to $keep_going if $keep_going is false"
  $keep_going ||= $job_B =~ /Out_$tag\.txt$/;
  next unless $keep_going;

  ...
}
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