Doc C
Minister of Fire
I'm an industrial engineer and not a stove expert by any means. A lot of this can be explained by simple physics. All chimneys may have some downdraft when stove is not in use. This depends chimney location (and height) in relation to the house structure, outside wind and wind direction, any open windows, HVAC system recirculating air inside the house etc. This is normal. What's not normal is that you have smell caused by downdraft.
Adding an external air supply system will not reduce the downdraft but may relocate the smell to where the air intake will be. The air intake should be very tight, with no gaps to keep the smell from inside the house.
I find it a bit strange that you get smell from a stove when it's not in use. Perhaps it only smells when the stove is done burning and you simply let it cool down for a few days while there are some slow burning coals in the stove. This is probably caused by lack of updraft from the cold chimney. This can be fixed by installing a dual-wall chimney liner inside your existing chimney. This will keep the inner chimney pipe a lot warmer thus creating sufficient draft, even at lower exhaust temperature.
Adding cap on the chimney will help to keep the moisture out and possibly reduce downdraft. A properly shaped chimney cap will actually create additional draft (low pressure) in the chimney when wind blows across it. If the cap is missing, wind may create slightly positive pressure in the chimney thus increasing downdraft (Bernoulli's principle from middle school physics).
Smell from a chimney not being used is extremely common especially if it’s a masonry chimney. The creosote forms in the chimney and gets wet either from no cap or from humidity being absorbed into the creosote.
When the air comes down the chimney instead of up the smell is sucked into the house.
There are several ways to approach the problem but it seems the majority of people that try each and every fix still come back and say they can’t get rid of the smell.