Even though it may not be as financially profitable in the short-term, it will always pay off in the long run. Don't be afraid to let them know of your doctorly aspirations. Even though you don't have your medical degree yet, I assume you have been through the basic pre-med curriculum. This training would give you a definite advantage in understanding medical terms and procedures, even though you haven't yet worked in the field. Your training to date is what you need to leverage when you make contact with potential employers.
Most hospitals post positions internally before they are made generally known to the public.
But you'll still need to make direct contact, either by phone or in person. You might even be able to find a position at the teaching hospital attached to the medical school you want to get into. These hospitals usually have more difficulty attracting candidates due to lower than market wages, so your direct interest alone may be attractive to them. Remember, it never hurts to know someone on the inside. What hurts is to continually stand on the outside looking in.