It's not exactly a new book, so some of the unsolved problems listed in the book may now be solved. In any case, it's one of the few I know that help a younger student go into more depth in an area where there's still active research going on. It's a difficult subject where many of the theorems can be proved without resorting to higher mathematics.
I'd imagine that there are probably similar texts for some areas of number theory and game theory, but nothing springs to mind. Non-Euclidean geometry may also be an option if the students have already taken geometry, and there were some text books that I found at least partially accessible in high school.
The Mathematical Tourist is even more out-of-date by this time. Since it's really a survey of many areas, it doesn't really meet your need, but you may find it useful yourself for looking into other areas that may be accessible to your students.
Finally, contact your local mathematics and math education departments. The math education folks may have some good suggestions. Many mathematics departments also do some sort of outreach to high school students, so there may also be some faculty there who could offer ideas.
"Hey Ivan, check your six." -- Sidewinder missile jacket patch, showing a Sidewinder driving up the tail of a Russian Su-27