Seems like the only way this would work is if it's done by individuals. I don't know how the New York State Constitution would be amended to have the State obtain properties around the world.
http://www.raken.com/american_wealth/OTHER/newsletter/chronicle280202.asp
Representatives of the Fords, Rockefellers, Carnegies, Astors, Hearsts and other extremely wealthy families get together in New York City and decide to pool their resources and create an elite meritocracy of the wealthy. They mean well. Riches ==> good education ==> good governance ==> more wealth for everybody ==> good society, etc.
They start buying up individual islands in the Caribbean, South Pacific. They buy large tracts of land in S. Africa and the Mideast. When the Spanish-American War comes and goes they take the opportunity to obtain control of large tracts of land in Cuba and the Philippines.
The Ford University of Technology is established in Havana. The campus covers several square miles. Representatives encourage Castro to join a baseball team and Cuba does not go Communist.
The Rockefeller Anthropological Institute is established in Johannesburg and
Apartheid is less severe in the following decades.
The Astor University is in Beirut; The Astor College of Near Eastern Studies is in Jerusalem. As a result, US influence in the Mideast is more evenhanded?
The Hearst Journalism School is in Manila. Marcos and Aquino become journalists instead of politicians.
The Cornelius Vanderbilt University of Technology is in Lagos, and the president of the university ends up controlling Nigeria since nobody else can or will.
The Cornelius Vanderbilt University of Science, Technology and Anthropology is in Liberia, and the president of the university also ends up controlling Liberia.
The campuses of these universities cover dozens of square miles.
Maybe, under the influence of these wealthy families, state-funded universities and colleges have gigantic campuses in Havana, Nairobi, and so on, covering hundreds of square miles.
New York might also do something with US Territories and Possessions like American Samoa and the Virgin Islands.
Thomas Edison starts a movie studio in the Virgin Islands.
Hugh Hefner starts his Playboy Mansion in the Virgin Islands, which I mention just because of what people think of when they see the word "virgin"..
That's the closest I can come to meeting the terms of this challenge.