I'm not sure that the cold would give you a sore back, I expect it's because you're not used to using a hammock.
A couple of pals still have issues getting a good nights sleep in a hammock after a number of years, especially on the first night of a multi day trip.
My advice would be to persevere with it, get a couple more nights under your belt and see how you get on.
Did you manage to stay fairly level or did you slide down into the centre at all? Try and get your feet slightly elevated, only around 6 inches but it can really help settle you down. Another reason might be because of the hyper extension of your legs, try and put a cushion/spare clothes under your knees, I've heard this can give people a bad back but I've never experienced it myself.
A decent underblanket is a good investment for the colder months but at this time of year you should be okay with your mat. You'll feel the breeze come through around your sides but it's nothing to worry about in July.
You might want to try and change the tension on your hammock too, I've not tried a TW but aiming for a 30* angle on your suspension should give you the most comfort. A lot of folk go for the tight stretched method so it does work, I've found a looser hang to be better for me though.
Have a good read around
this site for some tips