There actually are a couple of brokers that seem to do used cars on a lease - ie two or three years old, but to be honest they don't seem to be much, or any cheaper than a good deal on a new car. I think the economics of this are complex (a bit like the unfathomable pricing of airline or train tickets) - and there are continual deals from manufacturers on new car leases.
The key is to start looking early (as I have learned to my irritation) and be ready to pounce when you spot the one you want.