This question began with the notion that health care lawyers are advocates for their clients who are in the hospital. That is the kind of work the person wants to do.
Health care law practice is not generally about advocacy for patients while in health care. Rather it is about dealing with the laws controlling health care issues, enforcement of policies, legal or internal, and so on.
There are individuals who engage in health care advocacy, but not generally as part of the legal profession. There is, to my knowledge, no specific license required to be a health care advocate. If this is a field that interests you, there are health care advocacy organizations you can look into for more information. The National Association of Healthcare Advocacy Consultants and the Alliance of Professional Health Advocates to name two.
To get into health care law, your best bet is to join a firm with a robust health care department. A lot of health care focused law firms no longer desire to hire associates right out of law school, they prefer to have attorneys who already have some training in the area. So it is paramount to gain experience elsewhere. For example, relevant positions within the government that would be attractive to a law firm.
If you are interested in advocacy you might consider looking into elder law or medical malpractice. Another option is to specifically go into government law with a focus on health care.
You could look to join a health care advocacy organization that desires to have an attorney on staff and/or volunteer to work with such an organization. If you are able to obtain internships during law school you could then take the connections you acquired and the knowledge you gained and use that to obtain a position with a firm practicing in the areas that would enable you to help those you seek to assist.
This organization does the kind of work you are seeking to do.. So I would contact the attorneys there for advice on how to build a career. It seems to be a network for various attorneys working together to utilize their various skill sets and to provide pro bono representation.