toilet paper - easily forgotten!
Apart from that, it's not very easy without trying to tell you to suck eggs etc.
One of those mini fridges would be a good idea but this time of year possibly not needed. Baby wipes, I'm a lorry driver and keep a pack in my bag all the time, unbelievably useful!
Something to cook on and cook in - and a little bottle of veg oil and spare fuel.
If you're in the camping and caravan club (or can get hold of one) their "big sites book" would be very useful.
A small reliable torch.
Apart from that, I'm sure you'll work it out for yourself, it's not as difficult as it seems once you get going. It's not very easy to say really because I don't know if most of your time's going to be in the car or in the hammock but really I think if you've got the kit to be in your hammock you'll be fine for in your car, though I'd avoid sleeping in the car if possible.
If you can, ordenance survey maps will also help when it comes to somewhere to stay for the night. To be honest, I think you'd find it easier to find somewhere to pitch a tent than you will to put a tarp and hammock up.