what low cost hotels in SJ, near the airport, offer parking while a guest travels to a different city?
I thought you meant San Jose Costa Rica. It didn't occur to me that you meant San Jose, California.
On occasion, we have rented a car to drive to the airport (we live about 80 miles from the airport), and that would work, or you could use Uber or one of those types of services.
I spent a good portion of my afternoon trying to find a hotel with parking for my car and an airport shuttle. My trip is in the summer for two weeks and I couldn't find a hotel that wasn't already booked. I was shocked. I usually stay at Homewood Suites by Hilton San Jose AirportSilicon Valley. They don't charge for parking. Some of the hotels charge $10-15 per day. I'm going to just use a long term parking lot. Good luck!
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I think that a rental car agency could give you a better deal. Some of them offer off site parking at affordable rates. Check Budget, National and Enterprise
We have always turned in our car to the rental agency before we flew off to another city. So I don't, of my own knowledge, know that this is possible. We have left luggage at a hotel - most of them will store luggage for you. But I've never tried leaving a car.