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In the list of orders, you can display your own column and values by embedding four hooks.

actionOrderGridDefinitionModifier, actionOrderGridQueryBuilderModifier, actionOrderGridDataModifier, actionAdminOrdersListingFieldsModifier.

You can define your changes in the hookActionAdminOrdersListingFieldsModifier.

E.g:

public function hookActionAdminOrdersListingFieldsModifier(array $params)
{

    if (isset($params['select'])) {
        $params['select'] .= ', DATE_FORMAT(DATE_ADD(op.`date_add`,INTERVAL 7 + IF((WEEKDAY(DATE_ADD(op.`date_add`, INTERVAL 7 DAY)) IN (5)), 2, 0) + IF((WEEKDAY(DATE_ADD(op.`date_add`, INTERVAL 7 DAY)) IN (6)), 1, 0) DAY), "%W, %e of %M") as `DayOfDelivery`';
    }

    if (isset($params['join'])) {
        $params['join'] .= 'LEFT JOIN ' . _DB_PREFIX_ . 'order_payment AS op ON (a.reference = op.order_reference)';
    }

    $params['fields']['DayOfDelivery']['callback'] = 'getIsDayOfDelivery';
    $params['fields']['DayOfDelivery']['callback_object'] = $this;

}

public function getIsDayOfDelivery($a, $order)
{
    if ($order['note']) {
        return '<span title="" data-toggle="tooltip" class="label-tooltip" data-original-title="'.$order['note'].'" data-html="true" data-placement="top"><i class="icon-truck"></i></span>';
    } else {
        return '<span title="" data-toggle="tooltip" class="label-tooltip" data-original-title="'.$order['DayOfDelivery'].'" data-html="true" data-placement="top"><i class="icon-truck"></i></span>';
    }
}

Or another way.
Capture the change in order status when it is written into the ps_order_payment table and fill your date into ps_orders and the DayOfDelivery column. Alternatively, override ./classes/order/Order.php and the addOrderPayment function.
Then you can see the data straight away in your new column.