Encourage Preventative Care
Private insurance plans, especially lower cost / higher deductible plans combined with a healthcare savings account can be modified to increase the use of preventative medicine.
Rather than not receiving any insurance benefit until the entire deductible is reached, these plans could be modified to cover a given percent (50%, 75%, 80%...) of a specified list of preventative medical services. This would encourage anyone with such a plan to get preventative medical screening when indicated. More traditional plans could offer lower co-pays for these services.