Seaford-Oyster Bay Bike Path ?.

I mean the actual Seaford-OBY Expressway doesn't go to Oyster Bay, yet that's what we call it, so why not the same for the bike path ?.

In all seriousness, the name Bethpage Bike Path works fine as A) The original section follows the BP Parkway for most of it's length and B) The extension starts at BP State Park, if you use the standard map nomenclature of a road (or path in this instance) "starts" in the south or west.

Or we could call the existing as the Bethpage and the new as the Trailview as that's the park it's technically part of.

Papa Tom Expressway sounds better.