There were other unique and specific factors in 1865 and the decades after that made a new industrial revolution more likely in the postwar era. As World War I and World War II would also lead to economic booms later, the Civil War poured a huge amount of cash for the time into American war manufacturers. It is estimated that there were 16,000 new millionaires at the end of the war. Technology - in both weapons and in manufacturing itself - also advanced at a much more rapid pace than it did during peacetime.
These two factors, large amounts of government spending and advancing technology, made an industrial revolution possible. That same cash, now in the hands of a select few who were very wealthy, was invested into the private sector in all sorts of new ventures. Chicago and New York exploded in population and size, and the Westward Movement was stimulated by ease of transportation and manufactured goods.
In my opinion, the US was well situated to have an industrial revolution at this time because it was already a fairly advanced country but it, at the same time, had a number of factors that were going to allow it to grow.
The major factor that would allow growth was the fact that there was all the territory in the West to be settled. This area was more open to settlement after the Civil War because there were more railroads and because there were no longer arguments over slavery that would make it harder to organize new territories and states.
Along with the new territory, there were a couple of relatively new technologies that would open the territory up. In particular, railroads and telegraphs would play a huge role in making all of this area open to settlement.
With all this new land, the US had access to more resources than ever before. This would allow the country to experience an industrial boom.