Recover save data from a removed hard drive?

Months ago, I upgraded to a 2TB hard drive and made sure to upload all my saved data to the cloud so nothing was lost. However, I forgot to download one particular save file, and now that file has been overwritten in the cloud and I lost all my progress. I still have my old hard drive with the original file on it, but I know that it will be reformatted if I put it back in my system. So is there any way for me to extract that old save file and use it on my new drive or am I screwed?