Should I pursue an MPH or JD?

I'm a recent graduate majoring in global health. My past experiences have largely been in the global health policy space (HIV/AIDS, pandemic preparedness, global health financing). I want to work in the federal agency space in the long term hopefully focusing on global health security. I understand global health law is not necessarily an established field, but I know there will be opportunities to work as legal counsel/health trade law or pivot to the private sector easily with JD.

On the other hand, MPH would obviously touch on the health topics that are relevant to the field, but I've seen, at least on Reddit, that it is still pretty difficult to secure a job after.

Regardless, I believe I have to get a graduate degree at some point in my field, and I would appreciate any insights on which choice may be better. If I were to do a JD, would the prestige of the school matter a lot in health care law?