Since no one is going into why you need a recone, Ill explain that for you.
Your basket is bolted to the motor, often times to get these bolts out you have to remove the current softparts off the basket. Which means cone, surround, dustcap, spiders, voice coil.
Granted some baskets allow you to remove them from the motor without undoing all that, BUT you still need to remove the parts from the old basket to put them on the new basket.
The high strength CA Glue that is used to bond the spiders to the baskets is not going to allow you to remove the spiders without damaging them, no matter how hard you try the parts will not be reusable.
If you damage the spiders anyway you got to replace them, in order to replace the spiders alone you would have to remove the coil from the cone just so you can can access to the voice coil former to slide the spiders on from the top side of the coil. But before you do that you would have to remove all that old glue off the voice coil, and its not worth all that effort and time to clean that coil off and hope you dont damage its windings, or leadout wires, or deform it. Your carbon cone should be reusable as well as the dustcap if its carbon fiber, and even polypropylene caps can be reused at that point but the time to clean them off vs a few bucks for a new one just not worth it, now thats completly different when your talking about more expensive composite and carbon fiber cones and dust caps. After your all said and done a new recone reusing your cf cone is not going to be much more expensive then trying to rig the speaker back together with a new basket and surround, and after its all said and done its like a brand new woofer