Oedipus takes steps to save the kingdom from the plague. Is there anything he could have done differently that would still have saved the kingdom?
No, in a word. The plague is brought about by Oedipus' own presence, and he doesn't realise it. The process of cleansing the city from the plague, of removing the dirt which ruins Thebes, is the process of removing Oedipus himself. And for Oedipus to come to the conclusion that he must be banished from the city, he must know his own history. And he doesn't realise that until he has properly investigated the plague.
"Oedipus" means "I think I know" or "swollen-footed". Oedipus thinks he knows the problem, but he actually must discover himself, discover that the feet on which he stands are the feet of the murderer. It's a quest of self-discovery.
Thebes' plague is brought about by Oedipus having killed his father and marrying his mother, so the plague wouldn't exist without Oedipus having done those things. So not doing those things defeats your question. All that there is for Oedipus not to do is to discover his own identity: and that, sadly, is the only way the plague is removed.
Hope it helps!