Others have accurately explained how to get the sort to work the way you want, but I thought this would also be a good opportunity to tell you about the eminently useful Data::Dumper module, which will let you eliminate that unsightly loop at the end of your code.

use Data::Dumper;
# ... your sort code here ...
print Dumper \@big_table_array;
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Data::Dumper is handy to have in your toolbox when it's time for this kind of inspection.

        $perlmonks{seattlejohn} = 'John Clyman';