Stomp out the cigarette and flush and kill the bird. I never smoked myself but participated in this ritual many times. Dogs were drooling, etc but they were staunch for sure.
Cigarettes had another use back then. If dogs didn't seem to be able to smell anything, lighting a cigarette and holding it near the ground was often telling. On bad scenting days, cigarette smoke would actually not rise but would suck straight down towards the ground. When you saw this, you knew you weren't going to point any birds. It could change though in minutes or an hour or two and repeating the test showed the smoke moving up and laterally.
I'm glad we don't all smoke now but that was pretty useful in that country!