I think the idea is that you can't be everywhere at once and can't save everyone on your own. Rest assured, another new hero will be right along to help whoever you can't.
Later in the game, when you discover time travel through Ouroboros, you can do the other contact arc.
Right now, for a low-level hero, Officer Fields has you fighting Vahzilok, which are tougher enemies and more likely to send you to the hospital. There are a few groups of them around Atlas Park, so you could try fighting them and see how it goes. Hope this helps.