Hi there,
I agree with Paul. You have an interesting profile and an improved GMAT score along with your leadership position at the nonprofit will only serve to strengthen it. The high 600s isn't a bad foundation and given your background, I would really hone in on improving your verbal score. Any improvement here will have disproportionate benefits to your overall percentile. It isn't critical for each section to be in a certain percentile. If you can get the overall score where you want it, your story can speak to any concerns raised if one component is lower than you'd like. From a career switch perspective, you have nothing to worry about. Your goal is both realistic and ambitious. You at least have some ties that can be leveraged to justify the change but it's hardly necessary. Others have sold far more stretching stories. Not to mention, it's a growing industry and schools definitely want candidates who are looking to take charge there!
From an improvement standpoint, beyond the non profit, you're kind of done at this point. The non profit gives you established trajectory and even though it will be a new position, it clearly shows progression. If you start something new at this point, it will look a little out of place. Additionally, as Paul mentioned, you really need to get a few accomplishments in this new position under your belt. You have to be able to speak to something either completed or in the works. You need impact or metrics. Does that make sense? It would be great to say in the few months you were there you reduced X by Y%. But it also works to say you developed an annual plan to reduce X by Y%. Not as impactful, but hey, you just got to the position!
From a school perspective, I think you're selling yourself a little short. You do have a significant gap but that can be explained. A slightly higher GMAT would really open up doors for you. Even 10 - 30 points. The 5 - 10 ranked schools are realistic stretches for you. 11 - 17/18 are fairly well aligned and beyond that you have your safeties. I think a lot of great programs have already been mentioned. Tuck has a fantastic MBA/MHA program and I think that's worth a look if you're interested. In addition, take a good look at Haas. They are building up the healthcare component and they have access to a ton of start ups in the area that are beginning to focus in that industry. Finally, UNC has a strong program with great placement rates.
Overall, you're in a good spot. Take the GMATs again and (ok, I don't know the split between the sections) focus on improving your verbal to get your total score/percentile up. In tandem, focus on your non profit role and crank out a few accomplishments. I think the rest of your profile provides a lot of unique experiences that could be pulled in to create both a great and memorable application.
Let me know if you have any questions!
Bhavik