How to manage a particular “order by” for get_search_query()?

You can use the parse_query action hook, and set the orderby and order paramenters. 

function my_mod_search($query) {
    if ($query->is_search()) {
        $query->query_vars['order'] = 'ASC';
        $query->query_vars['orderby'] = 'post_date';
    }
}

add_action('parse_query', 'my_mod_search');

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