We took our son off of our auto insurance when he left for Basic.
He is currently in holding waiting to class up for A school, but he gets a two-week break for Christmas and will be able to come home for the first time in ten months. I know he will want the freedom to drive while he is home, but when he returns to A School he will not be taking the car (too old and not worth the upkeep upon arrival). He isn't planning to purchase a car for quite a while.
Are there any auto insurance companies that offer this type of short-term insurance?
If you give him permission to drive your vehicle, then he is covered in the event of an accident. That is how we handled my then Sailor driving when on Leave and he was covered when an accident occurred.
Make sure that his driver's license is current before letting him drive. Some have not kept their license current.
https://www.dmv.org/insurance/when-another-driver-has-an-accident-i... tells more about that.
If he has a driver's license, he should have insurance. It is required by law in some jurisdictions.
You could add him back on your insurance. Talk to the insurance company and explain the situation. They can help with appropriate coverage, which probably won't be that expensive.
Alternatively, and this may be the better option, your Sailor can get his own insurance, even not being a regular driver. USAA has always treated us right, and they would know the best option for any military-related circumstance.