If you have substantial hardware changes, always do a new install of the OS. Basically, if you change the motherboard, you have to do a new OS install.
There's basically two ways you can do this, each with pros and cons:
(1) Just do a clean install of the OS on the SSD, and manually migrate all your user files (save games, Word documents, etc.) from the HDD to the SSD.
Pro: just doing a clean install of the OS on the SSD is easy
Con: migrating all your user files can be hard if they're scattered all over the place
(2) Back up your HDD (in case of mishap or disaster), install it in the new computer, then do a new install of the OS on top of the old OS, telling it NOT to overwrite the non-OS files on the drive. Then, clone the HDD to...