When Spike TV became Paramount Network, it set out to build a new brand that reached women as well as men, while developing television that could stand out in a crowded environment.

Eight months in, it appears to have addressed these challenges in “Yellowstone,” a drama starring Kevin Costner as the head of a ranching family jockeying for power in Montana. Ahead of its finale on Wednesday, “Yellowstone” has become the second-most watched TV series on ad-supported cable, which excludes shows on streaming services and premium...