Now, I do not know the name of the game, but I could have sworn that it was "Jack Nicklaus Golf" that contained a tournament/season mode. After each hole (or round) you would see your current score against everyone else in the tournment (which would be 20 or more).
Is there an expansion disk to JNG that gives the tourney? Or was it another golfing game? I tried a few here, but none of them I recall from way back when.