I would like to extend my previous answer. Miles does tell Winnie another story. A very short story, which occurs in chapter 17. Winnie and Miles both woke up early, so Miles offered to take Winnie out in the boat again to do some fishing. While on the pond, Miles tells Winnie a little bit about his wife and two children.
"Remember I told you I had two children?" he asked. "Well, one of 'em was a girl. I took her fishing, too." His face clouded then, and he shook his head. "Her name was Anna. Lord, how sweet she was, that child! It's queer to think she'd be close to eighty now, if she's even still alive. And my son—he'd be eighty-two."
Next Miles tells Winnie why he didn't have his family members drink from the spring. He then quickly transitions into echoing Tuck's thoughts about why it is important that nobody else knows about the spring. The last thing that Miles tells Winnie is that he plans to use his immortality to do something important for the world.
"Someday," said Miles, "I'll find a way to do something important."
He doesn't know what it will be, and he knows that it will be tough, because he has so little formal education.