Smoke, or steam?
Steam can be condensation being steamed out, or it can be coolant. Condensation happens, and is worse if you make short trips, where you don't let the engine reach operating temperatures. I've seen the oil cap mayo happen in cars used for short commutes, without a BHG. Coolant can come from a BHG, but can also come from cracks in the block, head, throttle body, or ISCV.
Smoke can come from poor oil circulation, where oil is left on a hot surface too long without being replaced. Some WILL come from the cylinder walls, and is in part what the PCV system is there to catch. It can also come from combustion via worn rings.
In any case, smoking "a lot" means trouble. Fix it, and put your PCV system back together right, as not having it certainly didn't help you.