There is, indeed, a dilemma in Jem's mind after the trial. As they are awaiting the verdict, many thoughts go through Jem's mind.
First, how can Atticus and Tom Robinson lose? In Jem's mind, the evidence was clear; Tom is innocent. Atticus reminds Jem that they live in a racist world. Words from a white man always beat words from a black man. Here is what Atticus says:
As you grow older, you’ll see white men cheat black men every day of your life, but let me tell you something and don’t you forget it— whenever a white man does that to a black man, no matter who he is, how rich he is, or how fine a family he comes from, that white man is trash.
Second, when Jem finds out that the punishment for rape is the death penalty, Jem wants Atticus to change the law. Atticus says that something like this will take time, and there are no guarantees. Moreover, this action won't help Tom now.
Third, Jem's mind is turning. He comes up with the idea that more good people should sit on juries, since the power of a verdict rests with them. Atticus explains that even this is hard, as people have conflicts of interest. Therefore, they do not want to sit on juries.
Finally, as the deliberations are going for a long time, Atticus says that there must be at least one person arguing that Tom is innocent. He said that it was Mr. Cunningham. This revelation shocks Jem and Scout. Atticus explains that when you win over a Cunningham, they are loyal.
In light of these points, Jem's dilemma concerns what good people can do to bring justice into a system where injustice reigns.