Mark is spot on about how to download EE 1.7. Thanks Mark!
As for EE 1.6.1, within the EE 1 line you can download only the latest version available, we do not make older versions available. With that said, you have the 1.6.1 files, so I suspect that 1.7 is the right path for you to try.
We do recommend either updating and making sure that works then moving; or the reverse, moving then making sure that works before updating. Not both at once because then if something goes wrong, it’s going to be far harder to track down the problem. You might consider upgrading on the old server before you move it to the new server (since it doesn’t work on the new server).