If its not the carb, it could also be the valve seats.
If the valve seats are worn, it will allow a little oil to drip into the combustion chamber while the engine is sitting, and if the engine sits overnight or for a few days it will drip enough oil to cause it to smoke for about 30 seconds when you first start the engine the next time.
I use to have an old Honda walkbehind mower that would do that. It would spew black smoke for about 30 seconds and then it would clear out. The problem was worn valve seats.
If thats the case, its really nothing to worry about.
Sounds to me like maybe its a Kohler thing. My father and my brother have both owned Craftsman garden tractors with Kohler engines and from day 1 they both would spew black smoke when you first start them up. My father ended up returning his Craftsman tractor and exchanging it for the same identical tractor and the new one had the same problem. Im definetly not a fan of Kohler engines...