Have someone else run the car, and while it's smoking, give it a sniff test. If has a sweet and citrus smell, and it's lingering white smoke, then it's definitely coolant being burned off, which is usually a dead canary for a bad head gasket.
If it smells "oily" and pungent, and the smoke is lingering bluish-white, then you've likely got a valve seal or other oil-related issue in the combustion chamber.
If it smells like "gas" and black soot comes pouring out of the pipe, it's running too rich and you've got a fuel/sensor/injector type issue.
Try not to breath too deeply, don't want you to suffocate now.

Since it keeps happening even when the engine is revving higher, I'm kind of leaning towards a fuel issue, but more specific info might help.