I have also put on an oversize chimney roof over my smoke stack and with it built on metal fabric to keep out chimney sweeps and other rodents, but when it rains heavily my fire box leaks also. It makes a musty smell the water mixing with the ashes.
I think that my shelf above the damper is not curved enough for the water to go behind the actual fire box. I can see it drip from the damper area. Very frustrating, but I have learned to just live with it.