Hybrid CrossfireX only works with another AMD Radeon card. The video cannot be faster than a Radeon HD 6670 I believe, other Hybrid XFire automatically turns itself off. Not sure if it works with weak Radeon HD 5xxx cards.
Just put the 8600GT away in an antistatic bag for a rainy day. A working card is still good to keep around for emergency repairs to a friends / family members computer.
Try your new computer just with the onboard graphics. If you need more then the 6670 card is the fastest that will Hybrid Crossfire. These cards are $100 or less in the USA and cheap anywhere else. They make a big difference to performance and are easy to install and configure. They don't even need serperate power leads.