Just hold the keys, not the values.

push @holder, $key;
...
my @holder_sorted = sort { $hash{$a} <=> $hash{$b} } @holder;
print("@hash{@holder_sorted}\n");
[download]

Update: I see that people are suggesting that hold both the key and the value. It's unnecessary to hold the value since you already have a convenient way of looking up the value by key: $hash{$key}. But if you want to hold both, might as well use a hash!

my %holder;
...
$holder{$key} = $value;
...
my @holder_sorted = sort { $holder{$a} <=> $holder{$b} } keys @holder;
print("@holder{@holder_sorted}\n");
[download]