Either one of the two coils in the humbucking pickup doesn't work properly or the switch is bad. When you split the pickup, you activate one good coil and cutting out the bad one. The two coils normally work in series when not splitting. To see if the coil or switch are bad, you'll need to check it with an ohm meter. It's less expensive to replace the switch than rewinding your pickup. Check solder joints to all connections and make sure you have followed the wiring instructions properly. Sometimes the fine hook up wire can hit another contact or short out to ground. Make sure you check everything before pulling your pickup apart.