There was no strategic reasoning behind the timing of the second bombing, if that is what you are asking. The original plan was to have one bomb ready to use right after August 3rd. Then another would be ready by August 10th. Weather conditions over Japan pushed the first bombing back to August 6th, and then weather forecasts had them accelerate the second bombing of August 10th. Timing decisions were made for operational (not strategic) reasons by the soldiers on Tinian island, where the bombs were deployed. (By "operational" I mean, "decisions based on executing the operation in an effective fashion, without attention to the larger-scale political goals" and by "strategic" I mean "decisions based on predicted or perceived political goals"; the difference is between how soldiers on the ground view an operation and how a higher-level strategist or politician might. In the case of the atomic bombings of Japan, almost all decisions were operational, not strategic, despite the narratives we tell about the "decision to use the bomb").

I go into a lot more detail on the timing on this post, but the second bombing was not the product of a high-level decision. It is not clear Truman even knew it was going to happen; he was certainly not consulted. I wrote an article in 2020 that goes into a lot of depth into the ways Truman was and was not in the loop on the atomic bombing details; this was something he was definitely not in the loop about.

There is a lot of after-the-fact rationalization that occurs about the second bombing, but it has nothing to do with how it was regarded at the time. What Japan did after Hiroshima was send scientists to verify it was an atomic bomb, but the second bombing mission had already begun before they had a chance to act upon the results of that investigation.

I don't know what your comment on the "supporting public works" but you cannot find this kind of information in official speeches and Congressional records; these tend to only carry propagandized versions of history, deployed for specific political purposes (like justifying the use of the atomic bombs). You have to look at the works of historians who have gone through the papers generated by the people at Tinian and places like that. Michael Gordin's Five Days in August is a good place for a lot of this information, and its sources are extensive. The relevant papers are in places like the National Archives, though some are now online.

The quick answer is that the 2nd bombing wasn't "warranted" necessarily due to any particular action or event from 7 to 9 August. In fact it was warranted through inaction. The Allies were looking for Japan to "unconditionally surrender" as written in the Potsdam Declaration in April that year, and they would remain incredibly determined for Japan to follow through with the conditions penned down, even if that meant a second nuke or a naval invasion of the island of Japan (See: Operation Olympic or Downfall). And somewhat predictably, Japan would refuse to surrender after the 6th August bombing of Hiroshima.

Truman's speech that same day would be the first public declaration to Japan and the rest of the world regarding the nuclear bombing. The actions of the Allies in the following days such as radio broadcasts, air-dropped leaflets and Truman's speech itself confirms that they were quite confident in using a second bomb given their consistent rhetoric threatening such.

On Japan's side, Radio Japan continued to reiterate they were never going to surrender to the public and President Suzuki in his press statements (the exact words I cannot find just yet) also shared the same sentiment. The Army and Navy remained equally steadfast. The Japan government did not meet to discuss surrender or further actions until August 9th. This silence (that had already been present since Potsdam) proved to the Allies that further action on their end had to be taken to convince Japan through force that they would accept no other form of surrender.

The primary group that decided on the date of 9th August for the second bombing was Project Alberta which would be another place to jump off from and read more about. I found this document quite enlightening (Page 284 onwards). It can be seen that the decision to drop Fat Boy was made barely a day after Hiroshima and originally was going to be dropped on the 11th of August, but was pushed forward.

Unfortunately I don't think there is that much room into investigating the 3 days between the bombings although perhaps if one was to delve a bit deeper there might be room for more discussion? The events after Nagasaki leading to the final surrender of Japan are rather dramatic though and an interesting area to read about so I'll personally recommend that!

Notable Readings: Richard B. Frank - Downfall: The End of the Imperial Japanese Empire