Though I doubt the overhead will have a significant impact, even with really large hashes, using each is indeed more optimized.
That, and another optimization I thought of after posting my code, would change the subroutine to:

sub DigThroughHashref {
    my $hr_data = shift;
    while (my ($key, $value) = each %{$hr_data}) {
        if (ref ($value) eq "HASH") {
            print ("Digging through [$key]:\n");
            DigThroughHashref($value, @_, $key);
        } else {
            print (join ('->', @_) . "->$key - $value\n");
        }
    } ## end while (my ($key, $value) ...
} ## end sub DigThroughHashref
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All occurrances of $_ are now replaced by $key, and all occurrances of $hr_data->{$_} are now replaced by $value.
Also, the seperate variable @keystack is now optimized out. Instead of pushing the to-be-parsed key explicitly onto the end of @keystack, it is now simply added to the arguments of the sub (which results in it being added to @_ inside the sub).