Old Cavalry horses wore a halter under the bridle. The halter lead line was carried back under the pommel of the saddle, loose, with the end hanging out beside the riders leg. Should you fall off your horse, another rider can ride up beside it, reach over to the saddle, grab the lead line and bring the horse back to you. Never tie up your horse with the reins of the bridle, no matter what you see on tv or in the movies. If the horse should spook he could get a cut tongue or worse.
Good luck and enjoy your horse.
Do you have a picture of a cavalry horse with the lead rope?