If the new case has room inside you can store the spares in it.
I built my own so I can store my old /extra hard drives right inside.
BUT--If you have a whole new computer...
I guess you'd have to wait until the warranty is expired as opening the case usually voids the warranty.
Seagate.com has a disk wizard program that can copy the whole system on the new drive - all workable just as it was on the old drive.
if you have some bugs or viruses this isn't want yo would use though.