Look at more than the front apron
The side soakers and rear detail are often hidden from ground level. Debris can also collect behind a stack.
The masonry may be the route
Open pointing, cracked flaunching and porous brickwork can admit water above otherwise sound leadwork.
Unused flues can remain damp
A closed flue with poor ventilation may produce condensation, especially in cold weather. Damp on the chimney breast is not automatic proof that rain is passing through the roof.