Try turning off all accessories when starting the car. (ac, heat, radio, headlights etc..) See if that helps.
It could be an electrical type issue...battery or alternator perhaps.
Also on a separate occasion try leaving your headlights on for a few minutes before you start your car. Turn them off, then start up the car. This is supposed to be a good way to warm up tired batteries unless I am mistaken. If you notice a difference, that could be it.
There is also a cold start injector on supras that some people take off completely, since its not really neccessary in certain climates, and even in cold climates the car still starts pretty decent without it. You could check and replace that, but I have no experience with them so I dont know.
Also another problem I once had that was similar..
Check your fuel cap/fuel evap charcoal canister system. I don't really know if it is your problem, but it used to be where if I loosened the gas cap before I started it and hear air suck in or out, then leave it loose until the car was started... the car would start fine... That used to happen before on another car I had, not really sure what exactly was going on, but hey its something to try.
Other than that... I did have another problem similar to this before, where if I let the car sit overnight it would not want to start back up easily. It led me to believe that it was only when the car was cold, but it was infact only because I had let the car sit. It had a minor blown headgasket, and when the car sat, it would slowly leak coolant into the cylinders...then when you went to start it, it would sputter or stall, not really want to start.. and puff some white smoke out the back, then be fine. I really doubt thats your problem, but its one more possible cause.
either way good luck, let us know if you figure it out.