I've used Graham's posting to disassemble a 350 oil pump that I have, clean it out, retime it, and it now works beautifully. One of the things I've learned is that the black plastic pin that moves with the main gear will get stuck. I've been told by another site member that they usually get stuck in the wide open position, pumping oil as if the throttle is wide open. This may be the cause of many 350 owner complaints of excessive smoking. Anyway, I suppose it's possible for that pin to be stuck in the closed position too, metering out a small amount of oil. Since your pump hasn't been run in a while, it may just want a good cleaning and lubing!
For me the time spent following Graham's instructions were invaluable. Easy to understand and I came away with a great understanding of how the pump works.
Best of luck.