You can definitely work on teaching him to jog better on his back. You shouldn't need to go back to the ground.
To start, not all horses are built so that they can jog well.
It sounds like your horse is lacking balance and collection, which can be helped. He also may be locked up in his shoulders. It takes some time to build up the correct muscles and teach him to carry himself. It takes some patience, so don't expect an overnight fix. He also needs to get a bit more broke, which will come with working with him.
There's a couple of things to think about here.
How are you sitting on your horse? If you are tipping forward, you will be putting your weight onto his forehand, encouraging a fast, unbalanced trot. Ask a friend to video you walking and jogging your horse, and really look at your position.
I don't know what type of bit you are in, but ride two handed for these exercises.
Sit deep in your saddle as you'll use your seat to regulate his jog. Think about putting your shoulders slightly behind your hips. Keep your chin up and spine in line (through your neck on down). Craning your head forward is going to tip your weight forward.
Pull your leg back so that it's under you. A chair seat is going to get you out of position and not allow you to support your horse with your calf. Relax your thigh and allow the back of your thigh to be in contact with your horse. Keep your leg on him all the way down.
If you have experience using a martingale, it could help. If not, don't worry about it.
Starting at the walk, keep your weight back and legs on your horse -- not squeezing, but in contact with his sides. Keep light contact on the bit.
Start a circle. Sit deeper in your saddle, and lift your hands a bit to "block" him from rushing forward, and go ahead and squeeze in and back with your calf and foot. If he wants to speed up, add an "easy, walk." When you feel him round his back, drop his head, and slow his gait, quietly release the pressure and tell him he's a good boy. If he is resisting while you hold pressure, add a few tugs with your hands. Once he gives, pet him and tell him he's a good boy.
Another thing you can do at the walk and jog is to take his head to the inside of the circle while keeping his body straight, holding, and releasing. Any time you touch his face, you should support with your legs to keep him from trying to go hollow. Allow him to go straight, then take his head to the outside and do the same. This will keep his focus on you and should encourage him to loosen up his shoulders as he'll be reaching "through" a little more with the opposite leg when you bend him.
Riding boxes and square corners help a lot to get a horse back off the forehand. Remember to have a plan before you get to the point where you want to turn. Throw down a bunch of cones in a box. As you approach the cone to make the turn, you'll need to make sure that you are sitting back and down, keeping your chin up.
You're going to need to push his front end around the corner while keeping him fairly straight through the body. To push him around the corner, you are going to need to use your entire outside leg, your hips, and your hand. Your inside leg is going to need to support his shoulder to keep him from falling onto the forehand coming around the corner.
As you come to the corner, make sure he is in the bridle and elevated in his shoulders. Look over your shoulder to the place you want to go. Keep your shoulders square and hands square.
To make the corner, you'll need to lift your inside hand and support with the inside leg. The outside hand should move slightly across the neck, to begin the turn. Your outside leg is going to push him around the corner.
He should cross over or nearly cross over in the front. As he gets more broke, you can do this one handed.
All of these can be done at the walk, jog, and lope. Remember that every time you go to his face, you need to support and go in with your leg. Eventually, it should take less and less hand to get the desired reaction.