There is really no indication that the second bomb (Nagasaki) was due to any strategic logic at all. They had two bombs ready to use in early August 1945, the order for use gave authority to use as many as they had available, and the decision to drop the second bomb was essentially made by the local people on Tinian. It is not clear at all that Truman even knew it was going to happen; it was not specifically approved, Truman was given no advance notice of it being used, etc. There was considerably more attention given to the first use of the atomic bomb (Truman didn't want it done until the Potsdam Conference was over). The people who made the bomb and set the schedules (e.g. General Groves) did not know that two bombs would "end the war" — they assumed their job was to drop as many atomic bombs as they had, and Groves thought it would take maybe 5 bombs to cause Japan to surrender.

This is a separate issue from whether members of the US cabinet thought the atomic bomb would serve as leverage with regard to the USSR in the postwar. Some of them (notably Byrnes) did. But this was just one of many reasons they wanted to use the bomb. The bomb was, in a sense, "overdetermined" — from the perspective of Byrnes, Stimson, Groves, Truman, etc., there were many reasons to use the bomb and virtually no strong reasons not to use it.

But on Nagasaki — there was little high-level strategy one way or another. Which is not quite the revisionist argument, nor is it the orthodox argument. I have written a bit on this here and more specifically about the Nagasaki bombing itself here.