When my headgasket failed I never saw signs of coolant in the oil. Just coming out the tail pipe and signs of combustion gases and oil in the coolant recovery tank.
I think the key is that when we mean lots of white smoke , we're talking lots of white smoke. Like a cloud of white smoke. So if you're just seeing a puff, could be anything.
For sure you should see some change in coolant level if you're burning coolant. Either it'll go up, or it'll go down. usually down, but in my last case, it went up.
And yes, if the rings are having trouble seating, you can see some smoke due to that for a while. Local guy had a 4.6 block built forged and was blowing all sorts of smoke at a dyno tune. Took a good couple months to stop smoking. I'm considering the expensive dry compound from total seal to help the rings seat on my engine rebuild.