Short answer: It wasn't ready to use. Long answer: It's an issue of timing, orders, and technical limitations.

Truman authorized using the atomic bomb anytime after August 3rd, when the Potsdam Conference ended. The first day with an adequate weather forecast was August 6th, when the bombing of Hiroshima happened. The bombing order given by Groves, however, stated that more bombs could be dropped at will, and the next good weather day was August 9th, when the Nagasaki bomb was dropped. (It is not clear that Truman knew that multiple weapons would be dropped.) In any case, on August 10th, there were rumors that the Japanese high command had made a decision to sue for peace. Truman issued a "stop" on all atomic bombings — he didn't want any further bombings to happen without his personal approval, and it isn't clear he wanted any more atomic bombings at all.

There was then a series of negotiations between the US and the Japanese about surrender, where the Japanese attempted to secure some guarantees as to the status of the Emperor. In order to hurry these along, Truman ordered continued conventional bombing after August 12th. This was put on hold until August 14th. Around this time there was then an attempted coup against the Japanese Emperor, which was suppressed. The final surrender agreement was announced on August 15th.

The next atomic bomb would not have been ready to drop until August 18th or 19th. So there was no chance of it being used in this period. However the US commanders kept abreast of the schedule of the next atomic bomb, and General Marshall was looking into it as late as August 13th. But the plutonium core for the third bomb was not ready, and had not been shipped to Tinian, where the bombers would have launched from. (And even after arriving at Tinian, atomic bomb assembly was still a laborious and difficult procedure.)

So there was a strict technical reason that no third bomb was used — it wasn't ready — coupled with a stay on atomic bombing imposed by the President.

More details on the technical aspects of the third core here, along with discussions as to when they thought it was ready to use. Some discussion here on what they might have done with a third bomb if it was ready.