The rules say that players can bring additional characters into play at the beginning of the scene by paying a story token for each.

In the example of play, a character is brought in on a later page. I assume the example has it right, and you can bring in characters mid scene, right?

So if you can, do you have to do so at the start of a page? Or can you bring them in at any time on your turn?

Cool! Something else for the errata!

You can bring them in at the beginning of any Page. Not, however, in the middle of a page. There are some strategies that got used doing that in play-testing that disrupt the flow of play pretty brutally, so it got removed as an option.