The United States absolutely did NOT have a third atomic bomb available to drop on Japan, and this is a recorded fact, but don't expect to find it on the open internet. There's a long story to this, but there were basically two reasons as to why not; 1) the bombs dropped on Hiroshima and Nagasaki were of two completely different designs, and they wanted to see which one worked better before they committed to building a third, and 2) the 2 bombs dropped in Japan, plus the test bomb set off the month before in New Mexico had used up the complete supply of fissionable uranium available to make bombs.
The two bombs dropped in Japan were very different. The one dropped on Nagasaki was called Fat Man, and used an implosion system to create a uranium reaction. The one dropped on Hiroshima was called Little Boy and used a gun-type mechanism to fire a uranium pellet down a tube to create the uranium reaction. The advantage of Little Boy was you could make a less bulbous and more streamlined bomb that would more easily fit into bomb-bays. Fat Man however turned out to be more efficient explosively, by a factor of 10x.
Now in order to build and set off any atomic bomb, you need enriched uranium oxide. There was a special facility set up by the U.S, in Oak Ridge, Tennessee to do just that, but they weren't producing enough for more than about one bomb. Then a miracle happened. On the 10th of May, 1945, Germany surrendered and all German U-Boats everywhere were told to surface, raise a white flag, and proceed to an American, Canadian or British port. One of these German U-boats, the U-234, was carrying 1200 pounds of uranium oxide destined for Japan, so that Japan could start making her own atomic bomb. There were 2 Japanese officers on board the German U-boat to accompany this critical shipment, and when they found out the submarine was surrendering and headed for a U.S. port, they committed suicide. The submarine eventually pulled into Portsmouth, Virginia where the cargo was discovered. These 1200 pounds of uranium oxide now became US property and allowed the U.S. to make three atomic bombs, but only just. One was set off in the New Mexico desert in July as a test, and the other two went into the history books in August. If we had needed a third, it would have been at least a 3 or 4 month wait (and some say as long as 6 months) to get enough uranium oxide again, manufacture a 4th bomb, and ship it to the Pacific. There was a core available to build a third bomb, but that in itself was not the whole bomb. The core was intended for use in a third nuclear weapon, but after the Japanese surrendered, it was relegated for use in testing, It was designed with a small safety margin to ensure a successful explosion of the bomb. The device briefly went supercritical when it was accidentally placed in configurations during two separate experiments intended to guarantee the core was indeed close to the critical point. The incidents happened at the Los Alamos lab in 1945 and 1946, both of which resulted in the acute radiation poisoning and subsequent deaths of two scientists. After these incidents the spherical plutonium core that would have been used for this third bomb was referred to as the "demon core".
Ironically by the way, U-234 is a famous isotope component of uranium, and the fact the oxide was also on German U-boat U-234 is a pretty weird coincidence.