Perhaps I don't understand the question correctly, but the [very basic] time line is as follows:

July 26: The Western Allies issue the Potsdam Declaration - Western Allies call for the immediate surrender of Japan (among other things, like occupation and the roll back of the empire to the "home islands"). At the end of the Declaration, the Western Allies state:

We call upon the government of Japan to proclaim now the unconditional surrender of all Japanese armed forces, and to provide proper and adequate assurances of their good faith in such action. The alternative for Japan is prompt and utter destruction. (emphasis mine)

This last statement is important, as it is put the position of the Emperor into doubt. Unconditional suggests that the Western Allies would be able to do as they wished, and the removal of the Emperor was a non-starter among the Japanese.

July 28: Japanese publicly refuse to accept the Potsdam Declaration.

August 6: Early in the morning, Japanese time, the first Atomic bomb was dropped on Hiroshima. That evening, US time, Truman announced the dropping of the bomb and coupled it with Japan's direct refusal to accept the Potsdam Declaration:

It was to spare the Japanese people from utter destruction that the ultimatum of July 26 was issued at Potsdam. Their leaders promptly rejected that ultimatum. If they do not now accept our terms they may expect a rain of ruin from the air, the like of which has never been seen on this earth. (emphasis mine) Source

August 9: The Japanese had not yet accepted the Potsdam Declaration. The United States dropped the Nagasaki bomb, just as it had threatened to do three days prior. In a radio address that evening, President Truman again threatened total destruction of Japan if they did not surrender:

We shall continue to use it [Atomic bombs] until we completely destroy Japan's power to make war. Only a Japanese surrender will stop us. Source

For most of the day of August 9, the Japanese government contemplates surrender, but is unable to come to agreement. Late in the night, unable to agree, they consult the Emperor, who decides in favor of surrender. Early in the morning of the 10th, the Japanese send word to the Western Allies that:

The Japanese Government are ready to accept the terms enumerated in the joint declaration which was issued at Potsdam on July 26, 1945, by the heads of the Governments of the United States, Great Britain, and China, and later subscribed by the Soviet Government with the understanding that the said declaration does not comprise any demand which prejudices the prerogatives of His Majesty as a Sovereign Ruler. (emphasis mine) Source

August 11: The US replies to the Japanes offer of surrender:

With regard to the Japanese Government's message accepting the terms of the Potsdam proclamation but containing the statement, 'with the understanding that the said declaration does not comprise any demand which prejudices the prerogatives of His Majesty as a sovereign ruler,' our position is as follows:

From the moment of surrender the authority of the Emperor and the Japanese Government to rule the state shall be subject to the Supreme Commander of the Allied powers who will take such steps as he deems proper to effectuate the surrender terms.
The Emperor will be required to authorize and ensure the signature by the Government of Japan and the Japanese Imperial General Headquarters of the surrender terms necessary to carry out the provisions of the Potsdam Declaration, and shall issue his commands to all the Japanese military, naval and air authorities and to all the forces under their control wherever located to cease active operations and to surrender their arms, and to issue such other orders as the Supreme Commander may require to give effect to the surrender terms. ...
The ultimate form of government of Japan shall, in accordance with the Potsdam Declaration, be established by the freely expressed will of the Japanese people. Source

August 15: Formal Surrender of Japan, including the radio broadcast of the Emperor announcing such to the Japanese public.

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So to get to your question, I think there is an issue with the idea of "end[ing] the feud in a single sweep" (emphasis mine). The goal was to end the war, yes, but as far as I know, there was never the intention to only drop one bomb no matter what. Sure, if one bomb did the job, great, but better to be certain by dropping two (and, importantly, proving to the Japanese that the atomic bomb was not a one-off threat) and forcing the surrender immediately than dropping only one and having the Japanese believe they could weather the storm.

Does this answer your question?

It is tempting to read a "strategy" into the use of the two bombs, but there is little evidence of it in the historical record. It was a coincidence that they had two bombs ready to use in the first week of August 1945. The military order to use the bombs gave attention to the timing of the first bomb and then essentially wrote a blank check with regards to future bombs. It is clear that Groves, who was perhaps the most important person regarding the planning of the bombings, thought that at least two, perhaps as many as four or five, would be necessary to end the war, if any bombs would do so. The original planning was to drop the bombs a week apart (August 3rd and August 10th), but weather-related issues pushed the dates together (August 6th and August 9th). It is not clear Truman knew that two bombs would be dropped in such rapid succession; the only indication of a "schedule" that he was given regarded the schedule of plutonium bombs, and did not indicate that there was another bomb (the uranium bomb) ready to go. In any case, after the bombing of Nagasaki he immediately told the Army to stop using atomic bombs without his express permission — a sign, of sorts that he was reasserting control.

I have written a bit about historians' interpretations of the second bomb here, and on Truman's interesting actions (and perhaps misconceptions) here. It should be made clear, however, that even the way it did play out was not the way it was planned to play out.