I was reading a bit about all of the tremendous civil engineering taking place in Chicago, even before the fire, and I wanted to know how the screwjacks used were put into position in the first place. I couldn't find any firsthand sources outside of the Chicago Tribune archives, which tended to focus more on the Civil War at the time. Where should I look if I want to see the more technical aspects of how this was accomplished?
The site you found is actually about the most comprehensive research I've ever found on the effort (you did follow the links at the end, right?) It's possible there was a contemporary report in Scientific American; I don't think any of the engineering journals yet existed.
As for how the screwjacks were initially placed, remember that virtually any building in Chicago will have a basement, because once you've dug out footings for the walls that extend below the frost line, you may as well make the lower level useable space. So it was pretty simple in concept to carry the screwjacks into the basement, place them under the floor joists, and use a combination of cribbing and jacks to lift the buildings.
I doubt that there will be much scholarly work on the methods, as these would have been worked out on the fly. But I doubt that the techniques they used differ much from what we do today. You chop a bunch of holes in the foundation and insert cross beams. It is these beams that see the force of the jacks.
Here are some sites showing the recent moving of the Harriet Rees House that might give you a idea of the process.