You would use 'the next day' if this happened several days ago. But if she said it today, we would say tomorrow, because that is still tomorrow. Does that make sense? Here's another example: Last Thursday, my sister said she'd go to the post office to buy stamps for me the next day. I asked her about it today, and she admitted that she had forgotten to go. She said she'll go tomorrow. I hope she doesn't forget again!

It's a good question. I think you should. Otherwise, the reported speech has to take place on the same day as the direct speech, as you imply.

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