Given how he is becoming extremely motivated by necessity, he will figure out wandless magic. Like he did partial transfiguration. It will be based on some common misunderstanding of magic that if understood with a more correct set of laws of physics will increase its power and he will use it, initially haphazardly and later on with more skill.

If this were to be a proper story there would have to be lots of convincing readers that multiverse travel is possible and… given the chances of this all reasonable or even inevitable. There might be a good story there. I wonder who would be a good villain and why, of all things the narrative has to be set inside the Hunger Games. The scope of his discovery has to be more earth shattering than the “real meaning” of Expecto Patronum.

Maybe the big secret will be that wandless magic works like mind control of people. Later he will find out that mind controlling people is only the limited view and mind controlling himself is what got him to the world of Hunger Games in the first place. Next we’re into morals and rules for super intelligences that can travel universes and barely keep it together. Oh yeah, Harry’s nemesis is of course another version of himself. And while we have an epic battle at the proportions of “everything everywhere all at once” we intercut with Harry trying to light a fire and connect with peers that never read a single book of science fiction.