Billy, as did most people of the era that carried handguns, would have used the Colt Single Action pistol in any number of calibers. .45 Long Colt was popular, as was .38. Sometimes, people would buy the same caliber pistol as the rifle they owned so that they could use the same ammo in each. He probably would have carried the 4 3/4" barrel version for ease of carry and quicker drawing from the holster. The Colt also came in 5 1/2" and 7" barrel lengths. A Colt with a 7" barrel is more accurate due to the barrel length, but is a long gun to carry on your hip or tucked into your belt or waistband of your trousers! It was the finest handgun you could own, and is still very popular today.
It is known that Billy was fond of the Double Action Colt Lightning, which was a smaller handgun that fit his hand better. The Lighting came in .38 caliber and it's brother, the Thunderer, came in .41 caliber. This handgun was also known as a 'self cocker,' since all you had to do to fire it was pull the trigger, like most modern handguns. The Colt Single Action had to have its hammer ****** back using your thumb, thus the term 'thumb buster,' and then you pulled the trigger to fire the pistol. Using a Double Action Lightning would have also meant that Billy could fire off the six rounds, or five if it was carried with the hammer on an empty chamber to prevent accidental firing, faster than using a Single Action.
Here are photos of the three barrel lengths for the Colt Single Action and a photo of the Colt Lightning.