Not sure what you mean by additional driver. Did you book a vacation package through Southwest?
The kiosks are mainly used to check-in and print your boarding passes (if you haven't already) and check your luggage.
If you are just using the kiosk to get your boarding pass and not checking luggage, then you enter your confirmation number, print your pass and head straight towards your gate.
If you are checking luggage, you will still need to go to the counter as the SWA attendants will need to put the stickers on your bag, and make sure your bag is not over weight.
If you booked a rental car and are adding a driver, as far as I'm aware, most rental car companies will require you to go to the counter as they will need to verify the 2nd driver's information.