Unless you re-bless the object returned from SUPER::new(), you'll not be able to add your own behavior. Here's what that might look like when coupled with a couple of readability aids.

... preamble as before
use strict;

my $_count = 0;      # count of instances created

sub new {
    my $package = shift;
    my $self = SUPER::new(@_);

    # The class we inherit from counts its instances.
    # Since we're really creating one of us, adjust
    # its count downward, then point the object at
    # our counter.
    ${$self->{_count}}--;
    $_count++;
    $self->{_count} = \$_count;

    bless $self, $package;
}
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Without that bless, any attempts in call member functions would cause the search for that function to begin at the parent (overridden) class. Not what you want if you're going to be adding your own behavior in a subclass.

(I assume that you've examined the class you're overridding, and that swaping out $self->{_count} won't have any unfortunate side-effects.)