Based on the fact that oil consumption has not increased at all, I would be looking at fuel injection related causes in addition to burning of oil. With diesel fuel it can be a little hard to tell oil from fuel smoke by smell (both are oils really
).
But your turbo explanation sounds plausible also. Turbo failures on the D24T are very rare even at high miles. Just don't see them often. Rare enough that it would normally be way down the bottom of the list for troubleshooting of any issue under most circumstances. But believe it or not, the fact that yours has been rebuilt actually makes it more of a suspect for potential trouble vs even a well used original turbo, rather than less, in my mind. So perhaps worth further investigation in your particular case.
I would expect oil usage from an internal leak to get worse rather than improve as the engine warms up and oil gets thinner. But it's also possible to imagine a clearance closing with higher temp as you describe.
Another explanation is that oil is being burned in the engine cylinders due to leakage from the turbo on the inlet side, or from valve guides, etc. When the engine is cold it's harder to fully combust the oil (or fuel) leading to smoke. A hot engine can burn any kind of fuel more cleanly so that could be why the smoke disappears when hot. The cause of the smoke might not have gone away, but it's being masked by the more efficient combustion in a hot cylinder.
All that said, with no noted increase in oil consumption, another possibility is worn fuel injectors resulting in smoke that could then also clear up on a fully hot engine for the reason mentioned above.
Unfortunately I think the most effective way to make progress on pinpointing this issue would be to start doing a little bit of invasive exploration. To rule out turbo issues, you could disconnect the exhaust downpipe from the turbo outlet and look for wetness from oil. Could do the same on intake side of turbo (although this is hard to do conclusively since oil will always be there in some quantity from the crankcase vent tube, unless you clean everything up then reroute the CCV into a temporary catch can for a while and see if oil still appears in the inlet tract even with the CCV gases absent).
Then in terms of injectors, pulling them out and bringing to a Bosch shop to have them pop-tested and assess pop pressures and spray pattern is usually not too costly. Sometimes even just doing this with their testing fluid will have the effect of cleaning and improving the injectors. If needed they can rebuild them with fresh nozzles.
One other possibly relevant question. Is your engine equipped with EGR and if so, do you know if the system is still functional?