I'm honestly not sure that Odysseus does change very much during his travels. It's true that, in the end, he does display prudence and patience while awaiting his opportunity to avenge the wrongs done to his home by the suitors. However, Odysseus has shown prudence and patience before, when necessary; these aren't really new qualities that he's developed. For example, when he's in Polyphemus's cave, he prudently realizes that, much as he wants to kill the monster, if he does so, he and his men will be trapped in the cave. He needed to discover a way to maim the monster without killing him so that the Cyclops could still move the boulder away from the door. Odysseus patiently awaits his chance to blind the monster, and he is successful in his plan. He also knows that he and his men cannot rush out the door when it is open, because the Cyclops will grab them. He patiently awaits the opportunity to leave, prudently tying himself and his men underneath Polyphemus's sheep rather than on top, where the monster's hand might feel them.
Likewise, prudence and patience are necessary when dealing with the suitors. If he rushes in, holding his sword aloft, the suitors could defeat Odysseus as a result of their greater numbers. Again, he must exercise these qualities—qualities he's had all along—because, if he doesn't, he could die. He is brash, and he does have bravado, and he hasn't always been prudent in all situations, but it is a characteristic he's possessed for a long time. Then, the manner in which he and Telemachus decimate the suitors' numbers surely gratifies that bravado, which he's always possessed as well.