If someone suspects they need non-emergency medical care but they have to self pay, do they have options?

Yes. Yes, get to a good urgent care. They can handle most non-emergent issues, at a fraction of the cost. If they can't help you they can give you some direction about how to proceed.
HEALTH CARE OPTIONS. On behalf of the medical community at large, I apologize for our broken healthcare system. With that being said, I hope that you find that you follow helpful which includes a list of free/low income clinics in your area: http://www.freeclinics.com/cit/ny-bronx.