You could simply hook the old drive up to any port on the motherboard other than SATA0 (that one's for the boot device) and then turn the system back on. When you press Windows+E it should open up My Computer and show both the boot drive and the old "storage drive" which will likely show up as the next available letter after the already installed hard drive(s) and optical drive(s).
Example: If you have a hard drive as drive C: and an optical drive as drive D: then the old storage drive will likely show up as drive E:
PS - if you have application(s) or programs installed on the old drive, they may call on registry values that aren't there on the new drive. So as USAFRet said, you may have problems with applications installed on the old drive. But if it's just data, then there's no problem with doing it this way.