It's possible that you may be able to file a lawsuit, depends on the circumstances of how you were injured. New York has special laws that apply to construction cases. The most beneficial one for workers is called the Scaffold Law. That law says that if you fall from a height or if you're struck from objects that fell from a height that should have been secured, then the building owner and the general contractor are responsible for your injury. You may have your comp claim, but you can also sue the building owner and general contractor in that circumstance.
There are other aspects of the labor laws in New York that also apply, so tripping in an obstructed work area, for example, may give rise to a claim as well. The law allows a worker to sue someone other than his employer and puts a duty on the building owner and the general contractor to comply with these regulations to provide a safe workplace.