The main issue is - how perl interprets:

%$self->{simple_hash}
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My attempt to explain the issue is likely to create more confusion, so I will leave that to others.

What you could use in recent perls (>=5.20) is the postfix style :

 $self->%{simple_hash}
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Working code:

use strict;
use warnings;

{package toy_module;

sub new {
        my ($class, $args) = @_;
        my $self = {
                simple_hash => $args->{simple_hash},
                simple_test_scalar => $args->{simple_test_scalar},
        };
        print "********** INSIDE SUBROUTINE NEW **********\n";
        return bless $self, $class;
}

sub show_simple_hash {
        my $self = shift;
        print "********** INSIDE SUBROUTINE SHOW_SIMPLE_HASH *********
+*\n";
        print "simple scalar: ".$self->{simple_test_scalar}."\n";
        print "Simple Hash Key:$_\n" for keys %{$self->{simple_hash}};
}

1;
} # End of package toy_module

########## MAIN ###################
my $little_test_hashref = {
        COL_1 => 0,
        COL_2 => 1,
        COL_3 => 2,
        COL_4 => 3,
        COL_5 => 4,
};
my $test_object = toy_module->new({simple_hash=>$little_test_hashref, 
+simple_test_scalar=>5});
$test_object->show_simple_hash;
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