When I worked per diem for Flushing Hospital (which is partnered with JHMC and Brookdale), they ran an interfacility division. It's been a few years, but their EMS HR was located in the Axel Building near Jamaica Hospital. They handle hiring for all three hospitals. Just ask the EMT's or hospital staff where it is, and ask the guard inside where the EMS offices are. You can try to get on with their IFT division ad then move up to 911.
As far as hospital hiring, nepotism and favoritism in general seems to be dwindiling. When I applied to NSLIJ in 2002, I had family on the inside. They couldn't push my ppw through, but were able to influence the supervisor who interviewed me. Nowadays, more and more hospitals have an online application prgram, that is handled by the hospital's HR, which prevents favoritism until you interview, at least. Keep dropping off apps to the hospitals as well, and don't forget FDNY. even if you don't like the working conditions there, they look great on a resume.
At one point NSLIJ required 6 months paid 911 or five years vollie experience to do 911. This is from those in management that are active EMS volunteers. My point is that even they recognize that vollie experience can be questionable, and won't really give you an advantage in hiring.
Good luck with getting into a hospital, though, as you have plenty of unemployed, experienced 911 EMT's and medics out of work from the numerous hospital closings.
I'm glad I left NY when I did. Too expensive, the job situation sucks, and the quality of life is poor when compared to other areas. It's a great place to be from, a good place to party when you're young, but that's about it. Get your medic and NREMT-P, do some traveling to other states and maybe apply to third service or fire based EMS depts if you like the area. A wise man once told me "It's all about the best retirement." So true. When you start working 911 jobs and see how the elderly have to live, rotating meds due to costs, not being to afford heat/AC, or food, or ending up in a *****y nursing home, you'll begin to realize. You'll want a pension and continuing medical benefits post retirement. Some will say that you can do well if you plan properly with a 401k. The thing is, in addition to your pension, you can still do deferred comp, just not with any employer match in most places.
BTW, if you're pro-union, realize that there are quite a few right-to-work states down south and elsewhere. However, there's a bill in the Senate right now, that was already passed by the House, that will mandate collective bargaining nationwide for police, fire, and EMS. In NYC the non union hospitals have to compete with FDNY and the union hospitals to attract decent employees. I benefitted from this greatly at NSLIJ. In the south, however, if they tell you to jump, you had better say "how high, sir?" South Carolina was like that. They owned you. Northern Virginia has political "unions" (no collective bargaining yet, but effective via lobbying, campaign contributions/endorsements, etc.), so we don't get abused like they do further down south.