2001 Toyota Sienna burning oil

Our Sienna periodically belches white smoke at startup. But not at every start; it may go several days without smoking, and then emit a small puff, then the next day will emit a large cloud. If it smokes it generally does so only once on a given day, even if we start the car several times. I add oil once a week or so just to maintain the level.

Our mechanic, who we trust, performed a compression check but says he can't fiind "the " source of the problem. We feel we have to sell the car, and are concerned that it is now worth essentially nothing unless we can get it repaired.

Any experience with this problem and its resolution? Note that the car has only 57.5K miles on it, and has been well-maintained.