I just went 35 miles for the first time, and in the last 7 miles or so, my lower back was really starting to hurt. What's the best way to maintain proper back posture in order to avoid this in the future?
Ah. Well, you could have your bike fit professionally checked. You could go to the gym in the winter and do squats and back machines. You could do core work on an exercise ball. You could do pushups and crunches. You could ride rollers a lot, trying to perfect your spin - pedaling circles. You could ride more, riding shorter distances more consistently - like 6 days/week. You could go on a long hike in the mountains once a week. You can look at your shadow and check that your back is flat. You can imagine that a helium balloon is attached to the top of your helmet and is pulling your head and thus your back upward. Or you could do like me and do all of those. Back pain bye-bye, at least for the first 250 miles.
Yeah, check your positioning and build up the lower-back muscles. Keeping the back straight by rolling pelvis forward really helps. Imagine trying to place your belly-button on the top-tube of the bike, like curving your back so it bends downwards in the middle. Sticking out the belly really help you breath deeper as well.