Smoke coming from the engine

I put oil into my car today (the lights were showing that it was low on oil) and when I drove the car smoke started to come out of the engine.
There was no visible change to the performance of the car.

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Did the smoke occur before you added the oil? Which lamp(s) came on? Did the car stop running when the lamp(s) came on? Did you check the oil level before adding? How much oil did you add? Did you check the oil level after adding? Did the lamp(s) go out after you added the oil? Did you spill any of the oil?
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Put the oil cap back on??

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When you say 'came out of the engine', where do you mean? The tailpipe or the engine compartment?

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