Nice picnic area in montauk?

I've been in montauk before. I've been all the way to the lighthouse. Now it's all bare with no trees in the parking lot area to have a nice picnic. We're going with family and friends this weekend to see the sunrise. We'll be going there in the morning but I would like to find another picnic area. Does anyone know of a good spot that's not there??? Any other nice park area???