That was Chuck Connors second TV series, Branded. The Rifleman started each episode with Connors rapidly "fanning" shots from his custom, ring-levered Winchester 76, most likely into the carcas of some miscreant.
True story --- I dated Chuck Connors' niece years ago. His nephew, now a world famous deep sea fishing expert, is still a close friend of one of my brothers.
Connors' brother was a priest, one of those great priests of yesteryear who carried a Godly manner in a very masculine frame. He was a great athelete and just as handsome as his brother, maybe more so. He was just the kind of priest that has become exceedingly scarce in the Catholic Church today. As I see it, the Chuch's current troubles are directly traceable to that scarcity.