Yes, you can travel with your hedgie however you want...
However...You also have to think about the "what ifs". If there were ever an accident, or a need to do a hard break. You have your seat belt to keep you from flying forward, and in your shock, and sudden breaking, your hedgie will end up flying out of your hands, and off to who knows where(and suffering probable life threatening injuries in the process). Also, EMS are trained to look for pet carriers, not random piles of cloth, boxes, etc etc. So if you are ever knocked unconscious and unable to tell them there's a hedgehog in the car, your hedgie will be left in the wreck.
yes you can...but the question is ...should you? I don't think you should for the same reasons that Immortalia stated. Its just as unsafe for your hedgie to be in a car unprotected and unrestrained as it is for you.
As well as the safety reasons I can tell you from experience that it is difficult. The first few times I had to take Quigley on a car trip (I wasn't driving) I didn't have a carrier yet and had him in my lap. Once he got used to the motion he was squirming all over the place, peeing on me and generally being difficult. He gets to the point where I can barely keep him on my lap even when we are at home. I don't know how I did it those few times in the car.
I had a big accident with Mika and fortunately she's alive because she was in her cage. I rolled over with my car twice with Mika. At the end of that, she was still in the transportable cage. I though that she wasn't. I was sure that I didn't close her cage as I usualy do to be able to pat her.
It's really important to let any animal in secure in the car. Even if it a small ride!!
I feel very strongly that they should be in a hard sided pet carrier and ideally that it can be buckled in the seat. A small cat carrier with lots of fleece below and above will cushion the hedgehog in event of an accident.
I also can't imagine trying to keep an active hedgehog on my lap.
I too feel stongly about the carrier. After getting Litchi, I bought one because you never know what could happen. I always bucle mine up because, like Mika's accident prove, you never know what could happend.