You don't have to initialize hash members. Doing the following will work as long as $hash, $key, and $insert_value are initialized themselves:

#!/user/bin/env perl

use strict;
use warnings;
use Data::Dumper;

my %hash = (); 
my $key = 'A';
my $insert_value = 123;
push(@{$hash{$key}},$insert_value);

print Dumper(\%hash);
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Output:

$VAR1 = {
          'A' => [
                   123
                 ]
        };
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