There’s nothing saying that the horse will be terrified of dogs now. The horse I owned as a kid was attacked and bitten badly by a pack of 3 dogs and he didn’t have any kind of lingering fear.
One of my current horses was attacked twice, but not bitten, by dogs. Each time the dogs came running at her barking and snarling and grabbing for her hocks. She doesn’t have any fear of dogs now either. I have 3 dogs and mine don’t scare her at all, and subsequent dog meetings on trails has not bothered her.
The most important thing is to get the horse back into the presence of dogs, and don’t coo coo and coddle him. Stand up straight, hold the lead rope firm and with confidence, and move around the dog(s) as if nothing ever happened. If YOU get scared, then the horse will surely think dogs are something to be feared.
Horses have horrible trailering accidents and get right back in. A lady I used to ride with had her mare fall through a rotted wooden bridge and it took hours to get her out. She had no lingering fear of bridges and walks right over them like it never happened. There was a girl where I used to board who rode her horse down into a creek for a drink and the mare sunk in quicksand. It took four hours, about 10 people, and a truck with a winch to pull her out. She never had any fears of creeks or trucks or anything. She wasn’t even afraid to go back that same spot on the trail. She acted like it didn’t happen.
As for what season it is - don’t know about there, but around here people get their dogs in the woods all year long. Rabbits and squirrels are always open to hunting, so it doesn’t have to be any particular “season.” I meet dogs on trails all year long.
For what it’s worth - I have met PLENTY of unsocialized, untrained, unleashed dogs on trails of all breeds and types. In one situation it was a Black Lab and the dog was barking and lunging so furiously I had to yell at the owner “Would you PLEASE get your hands on that dog and get him under control?!?!!” The owner glared at me like I was the enemy.
Last year we rode on New Year’s day and we came across a guy in a pickup truck driving down the trail with a bed full of hunting dogs. The dogs were going nuts but the owner did yell at them and they shut up. But if they had been a little more untrained, they could have jumped out and we’d have about 5 dogs on our heels.
I also had to dismount one time and give a “BAD DOG!” and step hard into the dog’s space to make him stop and turn for home. He came out of his driveway and followed us down the county highway. He followed for a couple hundred yards before I realized he was back there. The more I yelled at him, the more he wagged his tail and thought it was fun. So I had to get off and get right in his space with a mean command and body language. Then he was like - oh crap - and turned and ran for home. I watched to make sure he turned into his drive, which he did. There was a car coming, and I flagged them to slow down - which they did. They crept along beside the dog until he went into his driveway.
We ride on two different public trails here that have lots of dog walkers and bikers. You see it all if you ride out away from home enough.