After reading The Just City, I was interested enough to continue the trilogy. Book two, The Philosopher Kings, was easily obtained from the local library.
Most of the viewpoint characters remain the same, with one major exception. Simmea dies (not really a spoiler since it happens at the very beginning) and her place is taken by Arete, Simmea's daughter by Apollo. I thought that Walton did an especially good job with Apollo's reactions to Simmea's death, through grief and revenge and acceptance. As a god, he's not used to that kind of loss, but is forced to deal with it in his incarnate form.
The Philosopher Kings feels more active than The Just City. The first book took place almost entirely within the one city, while the second deals with the divided island as well as a fairly extensive sea voyage. Until the very end of The Just City, there was very little large-scale conflict (though plenty between individuals). That's not the case in The Philosopher Kings, which starts with "art raid" battles between cities on the island, and continues with both physical and political altercations with those from outside.
Religion plays a larger part in this book than the first. With multiple cities, there's a wider variety of ideas of all kinds. For example, a form of Christianity takes root in some places, despite being in a time before Christ. Some people try to tie everything together into a single belief system, others permit free exercise to all faiths, while others establish state religions and persecute heretics. I personally find it discouraging that some people feel it necessary to establish exclusive beliefs and force them on everyone else, but a quick look around the world makes it pretty clear that's the reality of human nature. I think this is a pretty realistic outcome of the scenario that Walton has established.
Like the first book, the power of the gods plays a major role in The Philosopher Kings. This shows up mostly in the form of Apollo's children, who find that they are "heroes" with various kinds of supernatural abilities thanks to their heritage. The process is explained in a logical manner that I thought made the idea pretty easy to accept. There's very little overt exercise of godly power until the very end, when concern over the impact of the time travelers on history results in very major changes for the entire population.
Anyone who enjoyed The Just City should certainly read The Philosopher Kings, and I am definitely planning to finish out the trilogy.