If you are still negotiating your relocation package, ask them to pay the broker fees for an apartment search. WE moved to NYC in 2011 and Mr. Gfunk's company paid for everything. Our broker (who was top notch - let me know if you want an intro, not that he'd remember me) figured out quickly what we wanted and drove us around to see apartments in a big black SUV. Realtors and apartment brokers in NYC are worse than sharks. It's a super cutthroat world.
We wound up at 215 E 96 St, which at the time he said was the nicest and newest building on the UES. Also, he raved about the landlord being the Related Group - he said they were by far the most reasonable landlord in the city. They even let us charge our rent without a fee. So many credit card points...
For reference, our initial rent was $3200 for a 700 sq ft 1 br/1 ba with in unit laundry. It's a full service building - the gym even has a pool, which is rare. We moved to Long Island City when the rent hit $3600 three years later*. SPeaking of which, you should also consider Long Island City. It's kind of a hot neighborhood now, but there are lots of new apartments being built. You're one or two stops on the 7, E, M, Q, R trains to east midtown. You will probably be more likely to find something in your price range in LIC...or maybe Astoria (tons of great restaurants and maybe an extra 15 mins on the subway from midtown).
IF you would like to avoid the broker fees, spend a day taking a walk in the neighborhood(s) you're interested in. Especially if you look at staffed high rises, you can often just walk in on a weekend and ask for the rental office, who are happy to show you available units and talk pricing. If you are looking at walkups (likely in your price range) then you might need a broker.
*we now pay half that for a 2700 sq foot house on 1/4 acre