What you can expect from private health insurance will depend a lot on what state you'll be living in during school.
While $360/month might seem like a lot, realistically, if that includes medical, dental, and vision, that's a good deal... probably with low co-pays and deductibles as well. Look at all of those wonderful boring details before making your final decision.
If you go outside the ACA marketplaces, you're likely looking at something similar in price but with less coverage and no dental and vision. Really cheap plans will probably be "catastrophic" plans that only cover big expensive emergencies. It is possible that your school offers a supplemental plan that will fill in some gaps if you do happen to find a decently priced private plan though, so you do want to check on that.
If you go through the ACA marketplaces, which you can do even though the enrollment period has ended because you'll be losing your coverage when you leave your job which is a qualifying event, you might qualify for a subsidy and find an affordable health insurance option. You might also get prompted to join Medicaid if your income is insufficient. Obviously, with current political ****, no one can tell you the future of anything guaranteed by the ACA though. Tread lightly.
Personally, next to having an employer based plan, I would go with the school's (after looking over their plan's details). If you don't have an expensive chronic health condition, that is usually the best bang for your buck.
