What happens to the tax on my old car if I get a new one?

My stepdad has just given me his old car to replace the busted up Corsa i've had since I passed my test in 2017.

I've done the registered keeper stuff and it is in my name but I still haven't gotten rid of my old car yet (i've found a buyer but not done any of the paperwork yet) I've just taxed the new car but wondered what happens with the tax on my old one - Obviously, while it is in my name, I will presumably be paying to tax two cars, but I expect it to be transferred within the week.

What happens to the tax on that? Do i get a discount on the remaining tax that I would have had left on it or is it just tough shit and I just have to pay whatever tax on my new ride?

Sorry everyone, this is all new stuff to me.