In the story I'm currently writing, the MC is travelling around to various places. I'd like to introduce a side character to travel with them, but the best time I can think of to introduce this character and integrate them into the story is about a third through the overall arc. Is this too late to introduce such an important character, or does it not really matter?
I'd say it depends more on the execution that anything else. A third through the arch doesn't sound too late. It's up to you to integrate them in the story/with the other characters in the time you'll have. Write considering this and see how it turns out would be my advice.
I'd say anywhere before the half-way point should be fine. However, it would be good if you can find a clever way to foreshadow their appearance e.g. through mentions by word of mouth, in newspaper articles, on TV news reports etc.
In other words, it's never too late unless your book is near the end and doesn't have a sequel. Obviously, it depends on the circumstances in the book. But introducing a character only a third of the way through the arc should be fine.
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