Please don't ever, ever ride in side reins. They are stupidly dangerous if the horse is being controlled from a mounted position, rather than from the ground.
In order to soften him, you need to build up his muscles so he can carry himself softly. Transitions, transitions, and more transitions. Either from gait to gait (walk to trot etc) or within the gaits (lengthen, shorten). Don't let him go more than 12 or 15 strides without asking for something to change, for a transition, or a change in direction.