Keep in mind as well how a horse travels in the canter
Another position from which you can ask for the turn on the haunches is from the back up (as opposed to from the turn on the forehand as part of the 3-part maneuver). Ask your horse gently to back a few steps then when you feel him sitting on his haunches, use the aforementioned cues to ask for the pivot on the haunches.
Remember: don't tip your horse's nose (at least not excessively) and remember that setting your horse up is only one part of the deal; positioning yourself is equally important (perhaps even moreso). For an idea, I took Cody (our new Paint) out for a trail ride and, out in the middle of some fields (no fence or wall), I asked him to pick up the canter. Each time I asked him to pick up the canter, I positioned my hips and shoulders according to which lead I wanted him to pick up. I used no other cues (he was on a loose rein, I kept my legs even) and we were on a straight line going up and down hills. Each and every time he picked up the lead I was "asking" for, just due to the position of my hips and shoulders! Our position up in that saddle makes a drastic difference to our horses.