Could there be a new way to run?

Not if you want to run away from a hungry carnivore on the savanna, but if you are running on an artificial level flat surface—perhaps there is.

There is a precedent for a different approach in an artificial setting. The Fosbury flop is a "back to the bar" high jump method that is better done in an artificial setting with a nice cushion than in a natural environment.

Lifting

I wonder if lifting ones feet more than necessary above the ground is inefficient when on a perfectly flat surface. You need to lift your feet and place them down in long grass or on a slope or where you are not certain of the surface. Have a look at a 4 legged animal running fast. The front feet lift and place down presumably to avoid a tumble but the hind legs do not lift as much.  The animal would waste energy lifting the rear legs when there is no need to because a rear leg trip is unlikely.

Bounding
I was watching a slow motion film of a lion bounding along. The front feet were projected in front of the place they would land and gathered relative rearward speed before they hit the ground. When they did hit they imparted forward momentum.  
Here is an example of imparting momentum by striking something with excess speed. Imagine you want to spin a bicycle wheel using two hands. Try "running" on the wheel with your hands. As you speed up you use a lot of energy just moving your arms at the speed of the wheel. Now try using a slower rhythm but "hitting" the wheel with sharp fast moving blows. I bet you get the wheel to turn faster with this method than with any other and use less energy doing it.

Funny Running
About 50 years ago as an overweight 12 year old I walked what I was told was a measured mile (4 times around an oval) and was surprised that fast walking took only about 12 minutes. Being overweight there was no way I could have run the mile in 12 minutes but perhaps with a sort of fast walk I could do a lot better still. I was not walking in a "race" correct way as both feet were off the ground at the same time.  Using stiff legs and big strides I found that I could speed up and that at a certain speed I covered a lot of ground with little effort. After a party one night I used a similar method and did well in a race down the street. I can still remember that my body was upright and not moving up and down, my stride was long, my legs stiff and each foot fall was a brief impact
(with some heel) that imparted forward momentum. Just like spinning a wheel with "hits" I made the ground speed by under me. It was not very tiring and I could feel the pressure of the wind balancing my effort.  It may just have been the party "spirit" but it was a memorable experience.

Peter Chamberlain