engine smoke

Have someone driving behind you when you get on it to see if the smoke is coming from under thcar in general or out of tailpipe(s). If not out of the car, then check around the engine to see if you have any oil film on the covers/engine near the PCV/breather, oil on the exhaust manifolds/headers, or any place else under the car. It is either leaking, or blowby past the rings and coming out of the crankcase breather/filler.

If out of the pipes, then make sure of the coloration. Greyish is carbon/rich burning. Oil is bluish-tinted white and pure white is coolant. Your description of white/grey is kinda indeterminant.