TOKYO — Late in December, Taro Kono, the foreign minister of Japan, was hospitalized to have a ureteral stone removed. That night, he appeared at a party for the 84th birthday of Emperor Akihito. The next day, he campaigned on behalf of a mayoral candidate on the southern island of Kyushu, almost 600 miles from Tokyo. A day later, on Christmas Eve, he flew to Tel Aviv to begin a five-day trip to the Middle East.

It was the kind of relentless schedule, well documented on social media, that brought to mind a candidate for higher office. Which is what Mr. Kono is — just one who doesn’t know when his race may start.

Last fall, Prime Minister Shinzo Abe won a big election victory that put him on track to become the country’s longest-serving leader in modern times. Mr. Abe is 63 years old and has indicated he has no imminent plans to retire: He wants to stay in office at least through the Tokyo Olympics in 2020, and the governing Liberal Democratic Party changed its rules last year to allow Mr. Abe to do so.

But a new generation of politicians is already positioning itself to succeed him. Mr. Kono, 55, is one of the most intriguing, bringing a maverick streak to Japan’s usually stodgy political world.