yes, the lateral movement, cones or poles to turn around and backing around cones and stuff will help. Another trick that worked for me in the 'olden days' was to do lots of ground work, the side passing, the turning on the forehand and kind end, you know, german-style ground training, with the horse in surcingle and bitting rig (start with head low to level, and sloowly raise over time once the horse is using his back muscles fully in all ground work first. Head set is the last finish to the job, in my training. Anyway, with the bitting rig/surcingle and long lead road or rein set and a dressage crop just for signaling, and tapping hindquarters to initiate movement (some ppl just use the end of a long lead rope, I was trained to use use the tip of the dressage whip to encourage the hind end to move laterally, etc. Once the horse pivots on the hind end, backs around cones or through L shaped poles and side passes perfectly on the ground, and uses that hind end, you will find a new level of control for your horse. I am sure you have already done this, but posting this train tip advice for benefit of any others stopping by this post. I used to do eventing and then later in life just driving, and now am a brand new enthusiast learning about mini driving and trying to fit up a team to do some fun things with!
(I am new to driving minis, btw, and I have learned a LOT of new things and new train tips by reading the posts in this wonderful forum. YAY for mini horse talk, it is like having tons of friends right near you regardless WHERE you live! Love this forum!!)