Invest in a driving halter (nose buckle) if you don’t already have one, leave it on over the headstall for the foreseeable future, this is another level of control when you hitch and unhitch, also if for any reason you need to unhitch on the fly, you can lead the horse more safely.
This means you should also have a lead rope safely snapped in and tied out of the way on the carriage. If you don’t want to get a $pare$ kit, check out bike bags since they usually have stuff that can attach to a driving vehicle (if you want something waterproof, yeti makes a great bag that is easy access and bass fishing suppliers generally have stuff that fits the bill).
Put a good knife/utility tool in that bag. May we never have to cut our harness… May we always have a knife handy if the need arises <-- driver’s prayer
Always have your phone securely on your person and make sure it is secured, they tend to fall out of jean pockets. One of our club members had a flip over accident on the road to some trails, the gator was unconscious, the driver was ok, but had to head the scared horse. The phone was safely attached to… the carriage.
Routinely practice long halts in various places, especially when you first start off. This is helpful if you get distracted while hitching up and forget something (sigh… It happens). You want your pony to think it is no big deal that we went 30 feet, mom cursed and then we stopped while she finagled with the harness. (Like I said, it happens lol). My hitching routine is to count hitching steps. For me it’s 3/3/3… 3 attachments on each side, 3 at the head (lines to bit, false martingale at girth D). At some point it becomes habit but I spent 2 years counting every time and I still count at competitions! I’ve NEVER missed anything when I count. I plead the 5th on other occasions.
ETA - I don’t know anyone who removes the headstall and lines last, but I’m not sure I’ve paid a ton of attention to lots of people, but still I’ve shared the aisles with plenty… Lines are generally the first thing off after the vehicle is unhitched and the horse is clear (easier with independent shafts). My routine is to unhitch from the shafts, undo traces from vehicle from the same side (Quick release shackles so this is easy). Driving halter on, nose unbuckled. Lines removed from bit, laid over the back. Headstall removed, halter buckled on nose, take lines/headstall to their spot. Breastplate/traces removed, then breeching and saddle. I’m my case I don’t unhook the breeching. I can see where it might be possible to put a collar over a headstall on a vse, but it’s less easy on a large pony/horse without bonking them in the blinkers.