It's her suggestion to "jazz" things up that leads to her rendition of the song, and on from that to a plan to save Jamie by attracting him to her location as word of her performance spreads through the highlands. Nothing like a musical meme to make an impact - in this case, a meme from centuries in the future, a new time-traveling twist in jazz improv.
The payoff is not (yet) Jamie, but Dougal proposing to Claire, as the only way she can survive with Jamie gone, and Claire turning that into one last desperate attempt to free Jamie from the redcoats, with the help of just four men, none of them Dougal. And that's where this episode ends.
Just two more episodes ahead this season, with more to come next season, and I should mention again that I haven't read the books, so in my world anything is possible at this point. But I don't think that Jamie will die, as interesting as it would be for a Dougal and Claire marriage. So that means that Claire will have to find a way to rescue Jamie, against all brutal odds.
Further, it's unlikely that Claire will do this in a way that will make her Black Jack's prisoner again, because we've already seen that. But how, then, will she do it? Possibly the season will end with Jamie still prisoner, but that wouldn't be very satisfying, either.
My prediction: her survival, and Jamie's, will again draw upon some surprising knowledge or talent she has a woman from the future. Since I'm not a time traveler myself, and haven't seen the two next episodes as well as not read the books, I'll eagerly be watching to see how this all plays out.
See also Outlander 1.1-3: The Hope of Time Travel ... Outlander 1.6: Outstanding ... Outlander 1.7: Tender Intertemporal Polygamy ...Outlander 1.8: The Other Side ... Outlander 1.9: Spanking Good ... Outlander 1.10: A Glimmer of Paradox ... Outlander 1.11: Vaccination and Time Travel ... Outlander 1.12: Black Jack's Progeny ... Outlander 1.13: Mother's Day
Sierra Waters series, #1, time travel