They would have had a third bomb ready to use around August 17th or so. And approximately 3.5 bombs per month after that. However upon being informed about that on August 10, Truman ordered than no further bombs were to be dropped on Japan without his express permission. Early on August 14th, it looked like Truman might have thought he was going to authorize a third bomb (but it would take about a week to be dropped; it only existed as a core in New Mexico at that point, not an assembled weapon in the Pacific), but the war ended before he could do that.

There was no concrete plan ever made for the third one. If Truman had authorized it, they would have come up with one. There were different philosophies among people who were thinking about it at the time, ranging from "use it on one of the targets already authorized" (Kokura or Niigata), "use it on Tokyo" (as a political gesture), and "save it and use them tactically for an invasion." But nothing was ever decided.

You can read an article I wrote about this for National Geographic History here (text here if you are paywalled out).