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If you're willing to move to Queens, depending on where, you can get a larger apartment for $2,000/mo. Before you get discouraged, Queens is only one stop from Grand Central Station on the subway (maybe 7 minutes' riding). That is Long Island City, Queens. You can go even further into Queens inside of 40 minutes.
You can rent a 2 floor house with a backyard in Rockaway Queens for that.
Thanks to all for the great advice - in case anyone's interested I went on a scouting mission to NYC a few weeks ago and having seen 20+ apartments in 3 frenetic days, I found a small 1BR on 89th St & Columbus. I'm hoping to fly out permanently in the first couple of weeks of July (in the last stages of the visa process now).