The front flashing is only one piece
The sides and rear of the chimney often cannot be seen from the pavement. Side soakers, cover flashings and the back gutter all need checking.
Look above the roof line
Open pointing, cracked flaunching and weathered masonry can admit rain independently of the lead. Repointing good lead will not cure a split flashing, and replacing lead will not cure porous brickwork.
Unused flues can confuse the diagnosis
A cold, poorly ventilated flue may become damp through condensation. If staining persists in dry cold weather, do not assume every mark is caused by rain.