Paul:
After the Black it might be best for those so inclined to play the Red Course at the same complex. The Blue and Green are also worth a play for those who may not know much about the other courses at the complex.
No doubt Tallgrass needs to be in the mix. You also have a stellart layout -- although I don't how good the conditioning has been rceently with the likes of Montauk Down State Park -- the course is within four (4) miles of tge famed lighthouse.
I like Spring Lake in Middle Island -- good 27-hole complex. Sunkern Meadow is also a decent alternative.
Anong the newer courses you also have Oyster Bay in Woodbury -- a TF design that is routed in and around a former estate. I am not a fan of LI National -- located in the Riverhead area -- it's a RTJ, Jr. layout and is a bit contrived and far removed from antyhing remotely calling itself a links.
Harbor Links is a Hurdzan / Fry located in the Sands Point area and it took, if memory serves, a former garbage site and turned it into a muni-owned layout. Again, it's no links by any measure.
Among the old time public layouts worth a play include Pine Hollow and Crab Meadow and Middle Island.