I’d add Il Buco, Maialino, Pepolino, I Trulli, and Il Corso to the list of good Italian places:
and the one I mentioned earlier:
Any apartment rental for less than 30 days in Manhattan is illegal. Silly law, I agree, but it is the law. Now, this will almost certainly not affect you, but it is worth knowing, since, for example, the chance that your prospective host will get a fine/targetted/told to stop by the landlord before you arrive is non-zero. The gov’t has been trying to crack down recently. Perhaps make a backup reservation at a hotel.
Also, Greenwich is in Connecticut. 
Agree with the recs for Peasant, Il Buco, and Maialino. Maialino is probably my favorite of the 3, although probably the least rustic in terms of atmosphere. They have a huge by the glass program and lots of great barolos witha few years on them for not crazy prices. Rubiroso is excellent as well, but totally different food (pizza and red sauce). If you find yourself in the theatre district, Don Antonio by Starita and Pizzarte both have good Neopolitan-style pizza.
Yup, your last post made me curious so I looked it up and sure enough NYC has laid down the law on it. But clearly there are plenty of owners ignoring this. I dunno, we’ll see. Not sure how much of a risk it really would be…
Also, again, I appreciate the nice list of Italian restaurant options. Do any of these disallow BYO, or have astronomical corkage fees? Me thinks of splurging a bit at Chambers St., bringing to a dinner or two.
On the ones that have been mentioned:
Peasant - $35 per bottle
Da Marcella - allows for a fee, but the only time that we byod was when a friend brought bottles and paid for our dinner
Maialino - $25 per bottle (one of the best corkage deal, with good Italian dining, in town)
I Trullli - I haven’t byo’d there but eaten lunch/brunch many a times and they allow free corkage if wine is purchased across the street at Italian Wine Merchant
Il Corso - many local offlines occur here, I’ve attended a few where corkage was free, but always somebody else making the arrangement
edited to add:
Marea - $50 per bottle (the last time I was there 1 year ago)
Not Italian at all, but the Modern bar Room is no corkage on Sundays and has great food.
Maialino is a really good suggestion.
+1. Instead take a daytrip up to Arthur Ave in the Bronx. You will love it and May is the perfect time to visit. You could see the whole area in about 3-4 hours including an awesome meal. Real Italian passion found here. You could ‘Search’ this site.
Of course if you’re going up to Arthur Ave. May is also a great time of year to visit the New York Botanical Gardens.
Of course! and The Cloisters!
lupa and esca are favorites of mine
lupa on a warm day with the front windows open is a great treat in the village
A big THANK YOU to the Maialino recommendations in this thread. My daughter and I spent our one night in NYC staying at the Gramercy Park Hotel and eating a wonderful dinner at Maialino. Many hours at dinner and then just moved to the restaurant bar finish the evening. Went through a number of nicely priced Barolo wines over the course of six or seven hours, with great food throughout the evening. Great place with a lot of great folks working there, too!! Again, THANKS!!
Il Corso is now $35 a bottle corkage.
Il Buco.
Just checked back in on this one. More great suggestions, thanks again!
Excellent idea Mike, thanks. I’ll see about fitting this into our schedule.
And if you do, you’ll be a 5 minute drive away from the NY Botanical Gardens. Grounds should be pretty spectacular in May.