The spillage sometimes occurs in a cold chimney, but only before the flue becomes warm enough to create sufficient draft and pull exhaust fumes all the way up through the chimney.
That’s common on exterior type chimneys with 3 walls exposed and one wall shared with the house. Interior chimneys (only top section exposed above the roof) rarely experience it because the entire chimney stack surrounded with house walls is already warm.
If you’re constantly getting smoke spillage into your house (even after operating wood stove for a few minutes) there might something inside the chimney flue preventing proper draft. I would suggest checking the flue from the top. It could be a bird / bird nest, rodents, debris from trees, or deteriorate chimney material blocking the flue.
Third issue could be a negative air pressure in your home. This is sometimes created by operating exhaust fan(s) inside a house (kitchen, bathrooms, whole house fan) that are capable of reversing chimney draft and creating backdraft.
However, such fan(s) would have to be quite powerful and operating constantly to create backdraft effect for an extended period of time.
Other conditions creating negative air pressure could be:
Another chimney or chimneys with open flues
Poorly sealed windows and doors
Open floor vents to the crawlspace beneath and crawlspace venting to exterior, etc.
However, if there were no significant changes in your house prior to this chimney backdrafting issue, I would start from chimney flue evaluation.