Move Gift Card to directly under Payment Info

To remove it from the checkout page try this code instead:

function remove_pw_gift_card_redeem_from_checkout() {
    global $pw_gift_cards_redeeming;
    remove_action( 'woocommerce_review_order_before_submit', array( $pw_gift_cards_redeeming, 'woocommerce_review_order_before_submit') );
}
add_action( 'woocommerce_init', 'remove_pw_gift_card_redeem_from_checkout', 30 );

Marking this thread as resolved.