Atticus knew that it was very unlikely that Tom would win his trial. When Jem was bewildered at the verdict, Atticus replies:
I don’t know, but they did it. They’ve done it before and they did it tonight and they’ll do it again and when they do it—seems that only children weep.
Atticus' next move was to get an appeal together. Earlier, in Chapter 9 when Atticus is talking to Jack, he implies that it will come to an appeal. Atticus intended to to all he could for Tom's defense, but he recognized the reality that a black man in 1930s Maycomb did not have a good shot at a fair trial, especially when a young white girl was the alleged victim.
Apart from that, Atticus just keeps on leading the ethical battle by example. When Bob Ewell spits on him at the post office, Atticus does not retaliate, arguing that if Bob needs to vent some of his frustrations on someone, it is better to be Atticus than Mayella or any of the Ewell children. The appeal was only a small glimmer of hope for Tom, but it is the next step in challenging the verdict of the case. Unfortunately, Tom is shot before such an appeal can take place. And one can hardly argue with Tom's impulse to run (if that is what he did) considering how unlikely it would be to overturn the verdict.