I ended up going through my company to officially upgrade my phone, and they sent me a new iPhone 6. I had the same call forwarding issue with this new phone until I called AT&T and had them activate the new SIM card that came with the new phone. The issue appeared to only occur with my old SIM in the new phone (while it continued to work in the old iPhone 5).
Now that the new SIM is active in the new iPhone 6, call forwarding works as expected. It appears to be something tying the SIM to the phone, so maybe it’s an AT&T issue? Anyway, good to know if someone else thinks it’s just the same using their old SIM in their new iPhone 6 on AT&T’s network.