If I were you, I'd already start making plans to get to know your clinical instructors and those who do your hospital work with. I'm not a US educated student/nurse, but based on what my CA USA nursing friends would say is that you have your possible employer's people of your situation and that you truly want to work in their hospital.
You have the greatest advantage over us foreign educated grads in that you get the higher priority over hiring and have a better chance to be seen on your work habits, your personality, etc.
You already know that most of your batchmates are probably establishing themselves to work in the same hospital you're thinking of, the main disadvantage you have is not being a PR or GC holder or have a SSN, so it makes an employer think very hard before going thru all the steps and hassles of your residency status and waiting and waiting and waiting, when any hospital can just hire one of your batchmates who didn't have all the issues you have to get hired right after graduation.
The hospital doesn't need to pay for the bulk of the expenses associated with getting you hired. Good luck to you and keep us posted so that future foreign students can find your benefits. If you don't find success, that's okay to, let us know so that it might help someone else in your situation and I'm sure there are hundreds if not times 10 in today's world.