Since the game uses an independent launcher, you could try copying over the entire game folder on your drive (e.g. mine is D:\EDLaunch), downloading the launcher/installer, choosing custom location to install to, point it to the EDLaunch folder and see if it just validates the files rather than downloading them.
The game files download from the ED servers is painfully slow - I have a gigabit connection and it took well over an hour to download the last (~3.5GB) patch.
The process I outlined above works with Origin games (BF1942, BF3, BF4, Wing Commander II, Dead Space, etc.) Perhaps it will work with Elite and save you a bunch of download time.
What system are you upgrading to?