In the post about Balance I mentioned looking for overstep in the walk. When the horse is reaching up under themselves with the hind legs and lifting their shoulders and moving forward freely, you will see the overstep. In Dressage, both ridden and driving, the free walk is worth double points so time spent developing the free walk is well spent and well worth the effort. To paraphrase Xenophon, "If you don't have the time to properly develop the walk, how will you have the time to properly develop anything else?"
If your horse isn't overstepping and showing a proper free walk, then how do you get it? Good question. In the lines you might start at the halt. Shift your position back towards the horses hip. Before asking the horse to move, visualize How you're going to do this. As you ask for the walk forward, you will use your body and voice to push the horse forward. Your inside rein will keep the horse on the circle. Your body will be much like your legs and slightly push the hip out enough to allow the horse to follow their nose and stay bent to the inside.
As you move the horse forward, remember to keep your hands soft and Allow them to go. If the horse breaks into a trot, and they might because a forward walk can be tough at first, close your hand and use your outside rein to ask for a half halt. All you are telling the horse is, "No that's not what we're doing here at the moment." Don't get mad at the horse, just ask and if you have to ask over and over, then you ask over and over. We're just going to walk today.
When asking for the extended free walk, it is not a bad idea to have a verbal command for it. I cluck to Kat with each step for the most part. The whip bounces off his hip each stride also. Timing helps as I cluck as the hind leg is picked up and swings forward. When it's in the air already, it's easier for the horse to reach forward a little more and extend the stride. In the lines it is easier to ask and push the inside hind forward each time it comes up, so as the inside hind leg comes up, push for a little more forward using your body and voice. As the hind legs start to reach forward a little more, the horse will start to lift their shoulders a little more and their center of balance will begin to move back a little which helps them step under themselves a little further.
Since the horse isn't used to doing this reaching walk, accept getting a few steps at a time in each direction. When you see the overstep and the horse is reaching up underneath themselves, be sure to make a big deal about it and verbally praise them for their effort. This tells the horse, This! This is what I'm asking of you. This pleases me! Making a big deal of it helps the horse realize what you want. How can they please us if we don't let them know What pleases us?
If you're working in an arena or an open area, turns and serpentines are a great way to keep the horse listening to what you want and where you want them to go. After getting a few strides of a forward free walk, allow the horse to relax a little and then ask for a few more. Since you know your horse better than I do, you will know when enough is enough. As you work your horse over the course of a week and a few weeks, they will develop the muscles and the muscle memory to make the free walk something they will be able to do and hold a little longer each time. It's like running or lifting weights for us. The more we do it, the faster/further we can run or the more weight we can lift/reps we can do. It takes time, but you can and will get there relatively quickly once you understand the process and the horse understands what you're asking for.
With winter weather, walking is a great way to work off their extra energy while still having a productive workout that doesn't make them a sweaty mess needing a bath and a few hours to dry. Working at the walk is also a great way to maintain your connection with your horse and give them exercise after an injury also, given the horse is cleared by your vet for walking.