Unless horse is clipped, wearing a cover while working is going to overheat him. Horse will be LOTS more sweaty when you finish, than if you just use him without the covering.
If he comes home damp or sweaty, use a cooler to let him dry. We do that, leave the horses long haired, no blankets for pasture or barn time.
A cooler is a big square of fabric, could be the fleece, acrylic, wool, that covers the horse from high on the neck, to top of tail, and hangs down on the sides below elbows. An inexpensive bed blanket from the store will work fine! Just has to be big enough to cover him well. You could sew on front strings or use a spring clamp to keep the front closed. We put surcingles of elastic on our horses, so they can't twitch the coolers crooked or off. Binder twine girth will work fine to help keep the cooler on straight.
I bought the King Size bed blankets, acrylic, for our coolers. Our horses are tall and long, fit them fine. Other horses might need smaller sizes, not as long-geared as ours. They wash like a dream, were quite inexpensive and have held up VERY well for at least 10 years. Wool is great to, just hard to keep nice and not shrink during washing. I NEED to wash ours after about 6 drying sessions, the coolers just STINK. So easy washing is really important to me! The fleeces are really nice, but it seems that EVERYTHING sticks to them, hard to get off. You could make a cooler out of discounted fabric, seam it down the center if not wide enough. If you buy "horse" coolers from a catalog, they do cost more. I figure the $9 price on bed blankets for 5 of them, about equaled what only I could by 2 smaller sized horse coolers for, on sale.
You leave the cooler on and the dampness will evaporate thru the material off the horse. Cooler keeps drafts off him. Our barn closes up fairly draft free, coolers are not for outside wear, horse needs supervision while wearing it. If horse is REALLY soaked, quite cold out, I will double layer 2 coolers on him, remove the top one when the outside looks "foggy", from moisture. Actually is really damp, so I pull that top one off, hang to dry, let horse continue wearing the other, DRY cooler until it also is totally wet on the outside. Usually by now, horse is pretty dry at the skin level, his hair is dry as well, so I can remove that cooler and hang it up to dry as well. At that point I usually will brush the horse to break apart any hair that got sweated down, so he is fluffy all over again. If you have an EXTREMELY hairy animal, like some ponies, old horses, a third cooler might be needed to continue pulling off moisture. Most will dry fine with just one cooler if your outside temps aren't below 25F.
Cooler drying your horse can take an hour or two, so plan on that time before putting him back out in the cold. I don't really want mine damp at all, in temps lower than 25F., going outside. We usually drive in the afternoon, cooler dry them, and by then it is feeding time when everyone comes into the barn for the night anyway. They all get fed, coolers off, and I say "goodnight".
Ours do work up a sweat most drives this time of year, wearing all that hair while working. They get a long warm-up and cool-down time, since we are paved road and have to walk to the dirt road. It helps, just doesn't get they fully dry by the time we get home.
A working horse, in natural hair, is PLENTY able to stay warm in cold weather. You don't usually pile on heavy layers to snow ski, you will overheat doing the exercise. Movement helps keep the horse warm, they put out a TREMENDOUS amount of heat. The standing still time is when they need coverage, after you get home.
Just because YOU are cold on the carriage or sleigh seat sitting still, doesn't mean the equine is the LEAST bit cold! He is moving, staying warm.