Well, I think you have to visit the pyramid every time you advance training or you will have a hole. For example, if you are trying to move a horse from traveling like a roller skate (novice work) to traveling like a roller blade (more advanced work where the horse engages its hind legs to travel under it’s center of gravity), you will have to refine your concepts of straightness. You will have to work on straightness. As an example of that, you might use an exercise like spiral in and out at novice level to get the horse more straight and balanced between inside leg and outside rein. You might use that same exercise at the higher levels to achieve the engagement of the hind legs for roller blade straightness, but add in some shoulder fore or haunches out or haunches in.
For me, with younger horses now, I am trying to make them even and supple on both sides. Both of my young horses are a little stiff on the left, which is common. This impacts the contact, the straightness, the bending/suppleness, impulsion and even rhythm, etc. I feel it is important to spend time here in order to build the blocks to advance. So I am doing suppling excercises like figure eights and serpentines, but paying attenion to bend and straighten and have their shoulders aligned and keep their midsections from bulging. If I don’t have these things on the straight line and circle line first, I can’t really have it moving laterally. Then I add forward and back exercises to try to retain that level of straightness with impulsion. Sometimes you have to do a little less on one side than the other, until it catches up. For example the shallow serpentine at trot. Maybe I don’t go all the way to X on the stiff side yet, but work my way there. You get a real test doing that loop in canter because they will break gait if they don’t have the basics yet on the stiffer side the same as the other side where they don’t break gait.
Sorry for the rambling, but yet the basics. Test and revisit them as you move the horse along. Start with basic exercises and when you have mastered those on both sides, add challenging elements.