I recently taught a group of talented and enthusiastic riders in the Seattle area, and remember one rider performing lovely half passes to the left, but having trouble achieving enough bend and suppleness to the right. Her horse stiffened, and she resorted to trying to push him sideways, losing the bend and suppleness.
I remembered an old exercise I used to practice and teach years ago, which did wonders for suppling the horse behind the saddle and creating more flexibility and bend. It was also great for coordination, reminding the rider to use both legs.
This exercise is intended for horses performing a minimum of solid third level work, showing reliable collection and balance.

Tracking right in collected trot, ride shoulder-in right on the long side from M to B. At B, turn right across the arena to E in Travers (haunches-in) right. At E, carefully turn left, maintaining the right bend and continuing in Renvers (haunches out) down the long side to K. Continue in Renvers through the corner and turn left onto the centerline, maintaining the renvers.
Once on the centerline, immediately straighten the horse, ride a few steps of right shoulder-fore, and proceed in half-pass to the right, from the centerline to the rail.

You will be surprised at how fluid the half pass becomes!

Keep in mind that these turns are fairly steep and require adequate collection and balance, and that each movement is performed in the same bend.
Most riders find that the half pass gets much better, especially since these more challenging turns and movements tend to supple the horse nicely, and prepare him for the bend required in the half pass. They also remind the rider to keep the inside leg on in the half-pass, which is challenging for many riders. Give it a try!
S.H.