My first thought has always been to just make it mandatory, or at the very least opt-out instead of opt-in. I realize those are both very controversial, so I wonder why we don't have a monetary incentive. i.e. We will pay $1000 for every organ of yours we take to a person you nominate.
The fact that no money is involved is a critical part of this system. It also is the fact that no money is involved in RECEIVING an organ.
This makes the system purely altruistic a removes the kind of questions of "how much should an organ cost", "could I accelerate the process paying more for an organ?", "could I sell my organs, or the organs for my just deceased relative for more money?" "Is this enough money for selling my organs?"
Eliminating the money from the equation makes this simple. You don't win or lose anything for donating. If you don't want to, is cool. But if you want to, is totally selfless. Which strangely, makes more people be cool with the default and donate more. Because the only thing involved is feeling generous and helping others, unless you have some sort of religious feeling or fear. But, again, nothing to earn.