Here is the ambulance that drove Oswald to the hospital. As you can see, it's just a glorified station wagon with a back seat and back doors.
So, was it necessary to fold down the back seat to accommodate the stretcher? Well, in any case, the driver and the attendant were too far removed; they could not have put any blanket on Oswald as they were driving. So, how did the blanket get on him? Don't tell me that Leavelle and the other detective put it on him. Just don't.