When I tie, we use a halter and neck rope with a braided in snap. Sorry, i CAN NOT make myself tie up a harnessed/hitched horse and walk away as the Amish do! We carry the halter and rope for emergency spares if we break down, need to lead horse without carriage. Or for unhitching and tying up at a picnic drive while humans eat lunch and tell stories! Ha ha. I find it better to carry them in the vehicle than have horse wear it, because of the weight and movement on moving horse. It really annoys me watching it swing and slide, along with desensitizing horse to rein signals over time.
Some riding clinics I took wanted horse responding to light rein signals, so weighted reins, rein snaps on bit, runnng martingale, were dicouraged because they "hung weight " on the bit, constantly moving mouthpiece on mouth bars uas you rode, at all gaits. Deadened the rein feel to horse mouth. I want horse “talking back” thru the reins when I drive or ride him, not ignoring me. He gathers himself if I take up rein, preparing for what I may ask. Takes the rein back when I loosen them “hunting for the bit” to find contact. I do not want him flinging himself one way or another, throwing his head with a light rein touch. To me, that horse has an untrained mouth, over-reaction to any bit touch. He is NOT light-mouthed. He won’t move into bit pressure, unable to be collected, give and bend at the poll. Does not know how yet.
So I would carry any extra parts in the vehicle, not hang them on horse. Trail riding I securely tie halter and neck rope behind the cantle on my western saddle. Do not want it coming loose in travel. Out of the way back there, don’t have to watch them moving, yet still handy if I need to tie horse up for some reason.