Might seem a bit long winded but here is what I do.
Copy the folder containing the game from the steam library on the old drive to the steam library on the new drive.
Tell Steam to delete the local files
Tell steam to install the game and select the folder on the new drive, it then discovers the files and confirms installed.
As I say, maybe not the quickest but I'm generally just moving one game at a time, as once I finish playing something I tend to move it from my SSD to mechanical drive.