The answer to the original question is yes. If you install on a second machine with the same installation code, it does not remember what state you used on the initial install, and you can get whatever state you wants. I tried this up to the part where I actually downloaded the second state, but it seemed not to care what state I picked.
The previous answers did not address your specific situation or question.
Now, it's not clear what happens if you are planning to use this approach on the same return to be able to prepare two different states, what will happen if you copy the tax file over to the new machine and it finds a different state installed. But if you just want to prepare a different return on the second computer, that should work fine.
For those who will claim, "no, it remembers the state you used on your first install", please explain how people resell their copy of TT after they've filed (which I do not do but have heard of) and don't have complaints from people when they go to download their state software.
