The typical pre-med course work includes about four semesters of general and organic chemistry, two semesters each of general biology and physics, often calculus and statistics, and some upper level biochemistry and genetics courses. This can be done alongside any major by pre-med students, but if you have not been taking them since the beginning of college, you would have to take them as a visiting student (preferably at a four year college from the point of view of medical schools) after graduation.
If you do go into politics and policy, there are still major policy problems in health care that may need solving, such as the cost and complexity of the health care system. The US spends by far the most per person on health care (including both government and private spending) but does not have better outcomes overall than other rich countries that spend significantly less per person. Perhaps that is a way that you can help lots of people with respect to health care.