You don't need to go out trail riding until you can keep horse from increasing gait when you don't want him to. A horse that moves up and refuses to listen to rider's cues, is a horse that is showing it, and not human is running the show.
And I have seen a rider end up with shattered knee cap because she couldn't control her horse, and the other horse kicked and caught her. You are at this point a danger to any other riders who are with you if you can't control your horse.
Secondly, if your horse is like this, on trails you at this point do not need to be doing anything but walking. That your horse would get so close to another horse tells me you have little to no control over him. That is something you need to work on at the barn. That is where tightening up your cues, and his responses needs to be, not on a trail where you endanger other riders, or inconvenience them while you "train" your horse.
You can also work on this by the simple act of leading, haltering and unhaltering. When you halter, do not immediately head out of stall, but give horse command to back, and make him do it. Then bring him forwards, and tell him to halt, and stand there for a minute or so, no petting either, just stand there. Same with dropping the halter when you turn him loose. Make him stand still, back and come forward, and again no petting/praising. When leading, alternate your pace, and make him follow it, slow, then faster and then in-between those two. Stop and make him just stand there, and don't pay any attention to him, just stand beside him facing forward.
In a herd, the lead horse doesn't praise him for doing what he is supposed to do, he does it because there are consequences for not doing it. Needs to be the same here.
And if you are riding in a snaffle bit, that is direct pressure, and it will be hard to get horse to come back to you, as you are pulling directly against his neck strength. You have to give and take to get him back to you instead of solidly pulling against your hands.
But basics are what is needed here, not trail riding at this point. Can your father work on him for you?
Horses make me a better person.