you make a horse's mouth softer by having better feel and timing in your hands, and by supporting your rein cues with your body cues (and vice versa).
So, when you ask for something with the rein, the more perceptive you are to the horse answering your request and the quicker you reward that effort by taking the pressure off, the more responsive your horse becomes. THAT is a soft mouthed horse; responsive.
You don't MAKE your horse do things by the rein, you make your horse think about doing what you want him to do, and then he does it. You signal things to the horse, then the horse obeys (in theory) . You don't haul him all over the place. He moves himself , according to your direction. So, your direction only has to be strong enough to get through to the mind of the horse.
The other thing is to be sure to not "hang on the face" or "ride the brakes" of the horse. Be sure that there's an "off" position of the rein contact, so that "on" means something.