Normally I would tell you yes you need a new copy. I mean one thing you can't get around is re-installing Windows, that you must do.
Retail boxed Windows you can always install to a new system.
Normally OEM Windows 10 especially the upgrade, ties its key to your hardware, meaning you need a new copy. However many of our users have had success re-installing to new hardware. At this point with no definitive answer, I'd say try it. Best case, it works, worst case you need to buy a new Win 10 copy.
Supposedly the Win 10 Anniversary update makes a change to the licensing which attaches the key to your Microsoft account. If thats the case then it will make the copy portable between machines and give us a definitive answer to this...