The first use of the atomic bombing was well-planned out and discussed at very high levels of the US government. The second use was not — it was done because another bomb was coincidentally ready at about the same time, and the weather conditions were right. The decision to drop said bomb was made by people local to the island of Tinian, and neither the Secretary of War nor the President were consulted or even alerted. The strike order that Truman was shown (written by Leslie Groves, the head of the bomb project, and approved by the Secretary of War) gave very specific instructions on the first use of the bomb (including the timing):

/1. The 509 Composite Group, 20th Air Force will deliver its first special bomb as soon as weather will permit visual bombing after about 3 August 1945 on one of the targets: Hiroshima, Kokura, Niigata and Nagasaki. To carry military and civilian scientific personnel from the War Department to observe and record the effects of the explosion of the bomb, additional aircraft will accompany the airplane carrying the bomb. The observing planes will stay several miles distant from the point of impact of the bomb.

But essentially wrote a "blank check" on further bombs:

/2. Additional bombs will be delivered on the above targets as soon as made ready by the project staff. Further instructions will be issued concerning targets other than those listed above.

There was no coordination on strategy for the second bombing. No one was asked, "is three days an adequate time to assess the Japanese response to the first bomb?" (It was not: the Japanese survey group sent to Hiroshima to confirm that it was an atomic bomb had just sent confirmation back on the evening of August 8th.) The role of the Soviet invasion of Manchuria was obviously not considered (the invasion was just beginning as the second bombing operation was beginning).

The bombing of Nagasaki has been justified in retrospect as some kind of "strategic" move (e.g., show them we have multiple bombs, etc.) but it is clear that this was not how anyone was thinking about it at the time. They had two bombs, they were ready to use them, they had the authority to use them, they did. I suspect Truman did not even realize there were two bombs ready for use in such a short period of time (he spoke and wrote of the atomic bomb as a singular item until he got news of the Nagasaki attack). It is noteworthy that when, on August 10th, he was told there would be yet another bomb ready in a week, he told the military that under no circumstances could they drop another bomb without his personal approval. He told his cabinet that "the thought of wiping out another 100,000 people was too horrible. He didn’t like the idea of killing, as he said, 'all those kids.'"" I suspect this was in part an effort to regain control over the process, realizing that he had delegated far too much out to the military. (This interpretation is resonant with Truman's later, postwar handling of atomic weapons as well — he never again gave the military any leeway on their use, and barely even gave them physical access to the weapons at all, for the rest of his presidency. In many ways this is his more important long-term legacy with regards to the atomic bomb — the assertion of absolute Presidential nuclear authority — than his actions during World War II, where he was a largely peripheral figure in the use of the bomb.)

It is just worth being wary of many of the justifications to use the bomb, especially the "standard" ones. These were put forward by project participants only after the fact, in response to growing unease and criticism. Historians have known for several decades now that they do not at all capture the character of the discussions going on at the time. There is still room for interpretation on several of these points, to be sure, but it is clear things were far "messier" than the "orthodox" versions of this story would let on (and it is unfortunate that they are often regarded as the "neutral" version with anything challenging them as the "revisionist" version — they should be instead be regarded as "the version that the people who dropped the bomb told once they started to be criticized," and understandably be seen as self-serving).

I have written on how historians have viewed the bombing of Nagasaki here, and if you are interested in more details about the haphazard mission (which was almost a complete failure, and as many people know Nagasaki was not even meant to be the target, and was not bombed correctly, either), you can read my article on it here. In general, most historians of the bomb today believe that Hiroshima might have been inevitable given the circumstances, but that Nagasaki was probably preventable and unnecessary. The Japanese high command had a cabinet meeting on the morning of August 9th to discuss the Soviet invasion that had begun the night before; they received the information about Nagasaki during the meeting. There is no evidence or reason to suspect that it changed the opinions they had about the war (where the Soviet invasion definitely did). Even then, it took five more days, and an attempted coup, before they accepted unconditional surrender.