Car smoking after oil change

Depending on how much oil was blown out, it could take a day or two to burn off all of the oil.

I would use an engine degreaser first. Follow the directions on the can. That way you can be sure. Cover the coil leads with something before spraying it with the water.

Oil spilt onto the exhaust should burn off after 20 minutes or so and if it keeps smoking from within the engine compartment, chances there should still be a leak somewhere or spilt oil is being held in recessed areas and slowly dripping onto the exhaust.

As Mike explained, get the engine cleaned and run it till operating temperature and the smoke should clear. Do not degrease the engine while it is still hot. Let engine cool down before doing so as degreasing a hot engine can cause a lot of problem.

If the smoking is fomr the exhaust, chances it might take forever to clear if oil had gotten into the intake system via the crankcase ventilation system due to the overfilling.

If the above are confirmed to be the cause, you have no problem driving the vehicle around.