Rhodes' follow-up, "Dark Sun: The Making Of The Hydrogen Bomb", basically continues the story where TMAB left off. The narrative gets a bit more fractured and factual, focused on the question of how to safeguard nuclear weapons and what their political/military/diplomatic purpose is. But it does wrap up the J. Robert Oppenheimer story.
Once one has read these two, I strongly recommend "American Prometheus: The Triumph and Tragedy of J. Robert Oppenheimer" [2] by Bird and Sherwin. Also extremely well written.
[1] https://www.amazon.com/Dark-Sun-Making-Hydrogen-Bomb/dp/0684...
[2] https://www.amazon.com/American-Prometheus-Triumph-Tragedy-O...
The book details the Soviet nuclear program, as well as the truly massive amount of espionage that was going on at the various locations of the Manhattan project.
[1] http://www.amazon.com/Dark-Sun-Making-Hydrogen-Bomb/dp/06848...
http://www.amazon.com/Dark-Sun-Making-Hydrogen-Bomb/dp/06848...
The author also has a book about the hydrogen bomb too:
https://www.amazon.com/Dark-Sun-Making-Hydrogen-Bomb/dp/0684...