Good for you! You're right that it's safer to have a header but many of us have to hitch up alone and as you've guessed, there are things you can do to make it safer.
1) Teach your horse an ironclad "Whoa, Stand." He does not get to move until you tell him to whether you're holding him or not, in or out of the cart, and even where he can see you or not!
2) Cross-tying your horse in a broad aisleway where you can drive straight out would be ideal. Barring that, tie them to a hitching rail or something else that physically blocks them from moving forward. (I'm willing to be overruled on this one, I know everyone has their own opinion.) If you tie to a wall or hitching post, make sure the horse knows how to step over calmly and will stop and stand again before moving off. No anxious leaping forward!
3) When putting on the harness, never let your straps dangle on the ground. Secure the girth immediately as you can have quite a spook if the horse shakes the saddle off while you're doing up the crupper. Wrap straps (if you have them) should be secured through the tugs, tug dee's or rein terrets and your holdback straps on the breeching should be doubled back through their own keepers or buckled. Once the breastcollar is on, traces go immediately over the back and secured through either the backstrap or crupper area. Thread the reins through the terrets and secure the excess under the backstrap. Then undo the halter, put it around the neck and bridle the horse. Attach the reins now, not after the cart is attached. Never unbridle a hitched horse! Cart goes on last and off first.
(It is perfectly correct and a bit safer to leave a halter on as Mary mentioned, but I know my guys find that uncomfortable under a full driving bridle. If I want to tie temporarily while bridled I'll either use a gullet strap with tie ring or find a thin rope halter to leave on under the bridle. Buckle-nose halters are nice as well because you can release them without ever taking it off the horse.)
4) Follow safe hitching routines. Do up the
traces first because if the horse should leap forward we don't want the shafts to crash down out of the tugs and end up around the horse's legs. Immediately move to the
breeching, because without it if the horse leaps forward the cart is going to come along and then smack him in the heels when he tries to stop.
Once those two are attached, you can be a little more leisurely in adjusting things and doing up the
wrap straps or overgirth buckles. Those will only come into play when someone sits in the cart or the horse makes a turn so those are the last parts to be secured after the much more critical traces and breeching.
5) If you are hitching out in the open without being tied up, keep the reins in your hands at all times so you can control a potential spook and correct misbehavior. The horse should not
need to be held in order to stand, but you should be holding him none-the-less. He is not allowed to graze, mooch, rub his nose or otherwise do anything but stand there and peer around with interest. This is part of driving discipline and goes back to safety! He can step over a rein if he's grazing, hook his bit on the shaft while rubbing his nose...you get the idea.
5) Once the horse is hitched, push the cart forward and back a little to check adjustments and then double-check all your buckles. It's best to hitch on the flat or a little uphill as that simulates being in draft and makes adjustments easy.
6) Always gather your reins in your hand, have your whip and be in complete control of your animal before mounting the cart. The most dangerous moments without a header are between releasing the ties and getting in the cart. (See Rule 1!) Spend a lot of time teaching your horse that he does not get to move forward the minute the leadline is unsnapped or you get in the cart. I know people who get in and then literally sit and read a book for ten minutes!
7) When you are out, tell someone where you are going and when you expect to be back. Carry a cell phone
on your body. It does no good if it gallops off with your runaway! Have a tag with that cell number attached to your horse's harness in case you get separated. Given where you live I'd carry bear or wasp spray, put a small bell on your harness to warn wildlife you're coming and if there's a hunting season near you I'd wear International Orange and put some on your horse. Vehicle reflectors are also a good idea.
I know #7 goes beyond what you asked, but safe harnessing and hitching is part of safe driving and I thought it had a place here.
I look forward to seeing what our other whips have to say!
Leia