As long as I have a rifle, or musket, I carry something suitable for small game. Either my S&W AirLite, or an Uberti 1862 "Pocket Police". Sometimes my Remington New Model Army in Navy caliber, and sometimes my 1860 Colt. The Remington is good on grouse, the 1860 I just plain love it.
I agree with those who carry a powerful revolver in case they get caught away from the rifle, but I try to minimize the weight I carry and depend on the long gun, and take my chances. When I was young and strong as a mule, I carried my 7.5" Super Blackhawk everywhere when hunting. Didn't bother me at all.
The Webley has gone with me on a few hunts. It's power to weight ratio is not great, but it works well with shot loads. And I love it.
Haven't shot a grouse with the Pocket Police yet, but she's got the accuracy to do it.
The AirLite has fixed me dinner. Carrying this is like not carrying a pistol. It's that light.
The Remington was my constant companion when hunting for quite a few years, and has taken quite a few grouse. Not a real heavy pistol, but these days I prefer the 1862 or AirLite due to their lighter weight.