A short length of rope is called a tie-down rope or a “piggin string” by cowboys. It has multiple uses and lengths and it a common item seen attached to saddles or sometimes chaps.
You’ve probably see one if you’ve ever watched tie-down roping. It’s the short length of rope that the cowboy holds in his teeth while he catches the calf and wrestles it to the ground. He then takes the pigging rope and wraps it around three of the calves feet and fastens it with a half-hitch. It’s sometimes called hogtying and in fact it can be used on a lot of animals to restrict their movement.
In real ranch situations you may also tie a cow’s feet that same way so you can doctor it or brand it although you would greatly prefer to do it other ways that immobilize the cow better.
They have a lot of other uses including helping pull fence posts together at gates so it can be closed, hobbling your horse, making a temporary halter, tying things onto the saddle or a pack horse rig, tying a loop around a tree that can be used to attach your lead rope to, tying up bad guys, etc. etc. A great contest would be to try and come up with the most uses.
These days most tie-down ropes are nylon or polyester and 1/4″ to 3/8″ in diameter. Some even come in colors. For ranch work most cowboys I know simply take an old piece of rope and cut it to somewhere between 5′ and 7′ just depending on personal preference.
Very handy thing to carry.