Contrary to opinions above, traffic WAS NOT any problem in 1995. In fact the traffic was MUCH lighter the weekend of The Open as the reports of traffic scared the typical weekenders and they stayed in NYC !! I drove two days from my house in Southampton and parked in the Shinnecock Indian reservation and took the bus, and I took the train one day (from the Southampton train station, see below). On Sunday, I had a wedding back in NYC and drove right through, no problem. Trust me, Shinnecock is much easier than Bethpage.
My suggestion is drive past Shinnecock (coming from NYC) into The Village of Southampton or Bridgehampton. Park at the Southampton (or Bridgehamton) train station or nearby and take the train EAST towards New York and get off at the Southampton College train station, which is right in front of Shinnecock. Normally not many trains stop there, but they all stop Open week.
After The Open, you take the train back to The Village, have dinner and then drive to your hotel traffic free. Also go check out Flying Point Beach in Southampton, it is beautiful.
There really are very few hotels in The Hamptons, so don't try a Internet service such as Hotels.com. Your best bed is a small hotel or B&B, see http://www.southamptonchamber.com/
for a list, also try a similar list for the Hampton Bays Chamber of Commerce.
Finally, there is a way to drive back to see National even though the obvious roads will be blocked off. It is very very complicated, but can be done via North Sea road.
PS. If traffic was as light as it was during Open week all the time, we probably would not have sold the house.