@kayakat The EA App should recognize the data already present and simply verify the install. Sometimes it does simply do that. Other times it downloads everything all over again. I haven't tested in a few months, but I can tell you this was highly unreliable when I was last experimenting—the App even verified some expansion packs while reinstalling others, when they were originally all installed together.
As a side note, is there a reason you're reinstalling Windows on an HDD rather than your SSD? Windows will be much more responsive running from an SSD, to the extent that it's like having a whole new computer. If you don't want to erase the SSD, you could instead partition it, leaving only enough space for Windows itself and the page file, plus room for updates; and redirecting user folders (Documents, Photos, Downloads, etc.) to the other side of the partition or to your HDD. 80 GB should be enough as long as you really do move everything else, or 100 GB to give yourself a larger margin of error.
