Q&A

An engine that is burning oil emits blue colored smoke that has an oder of burning oil. It is heavy blue to blue gray in color. If the engine has worn valve guides or valve guide seals oil may drip into the cylinders when the engine is turned off and allowed to cool, then when you start the engine up in the morning you get that trail of smoke out the exhaust. light colored smoke out the exhaust is normal until the engine warms up, especially noticeable on cold mornings dense air. White or sweet smelling smoke out the exhaust perhaps accompanied by rough running would be a sign of a blown head gasket.
Some oil consumption on a higher mileage engine is normal, Mercedes used to say 1 quart per 1000 miles before they would do anything under warentee, (which I feel is excessive).
If your car badly over heated and that was the cause of the head gasket failure, the overheating episode could have damaged the piston oil control rings and oil is now not being "scapped off" the cylinder walls by the oil control rings.
I would let the car cool down, start from cold allow to run fro a few minutes, remove the spark plugs see if they have oil fouling (not carbon fouling due to rich fuel mixture on cold start up). Then I would drive the car and get it up to operating temperature and do a cylinder leakage check. This checks the condition of the rings but cannot access the exact condition of each ring (either oil control or piston compression rings) it just gives you an indication of blow by past each piston.
Perhaps replace the PCV valve it is cheap and easy to fit. Does your car have any engine oil leaks and check the tailpipe for signs of oily deposits in the pipe itself.
I am afraid the route of your problem may still be internal in the engine.