For students who are graduating with an MPH, if you're interested in a career in hospital administrative medicine, then perhaps you'll want to pursue a hospital administrative fellowship program. For instance, the Massachusetts General Hospital has a great program and you can view a list of current and past fellows here.
Also, the Cleveland Clinic has a hospital administrative fellowship program and you can see what the former fellows are doing by visiting this link.
Hopkins also has an Administrative Fellowship Program. You can learn about it here. You'll notice a common theme here: the fellows have an MHA or MHSA or MPH degree. Also, most of these fellows are not physicians but they are graduate students who have a strong interest in hospital administration or health administration.