During grad school I was a student ambassador (or something like that) and I was tasked with helping foreign students here on study abroad for 5-6 months get situated. Cars were always a problem since there seemed to be a big gap between short term (a week or two) and long term leases (24 months minimum). I want to say they figured out it would be much cheaper to buy a car and then sell it when they were about to go home. The thousand or two dollars potentially lost on the sale was less than paying the cheapest rental rates for a month at a time. I want to say the best they found was $2000 for 3 months at Rent a Wreck or something, and I think it was an older car.
I would think you could get a 8-10 year old Honda or Toyota sedan for $5000 or so, drive it for 6 months, then sell it for about what you paid for it without too much trouble.
I think you can also ship cars from one place to another. Depending on mileage between origin/destination, that may be economical and more convenient than renting or buying and selling.
Another option, particularly if public transit is somewhat decent, would be to rely on occasional rentals and/or zip-car type hourly rentals.