Saddle scabbards began to appear in the 1870s. The US Cavalry did not adopt them until the 1890s.
The "saddle ring carbine" had a ring that was used to hold the carbine -- in military applications, a "carabiner" on a leather strap that ran diagonally across the chest clipped to the ring. Civilians mostly tied a leather thong through the ring and looped it over the saddle horn.
Randolph B. Marcy, in The Prairie Traveler described a piece of leather with two "button holes" in it. The leather lay in the rider's lap and was "buttoned" with the saddle horn. The rifle lay across the leather, and the other end of the leather was brought over the rifle and "buttoned" to the saddle horn. With such a rig, Marcy could ride with his rifle across his thighs and his hands free.