Be careful with prednisone tapering, most importantly, don't start it too soon. Stay at 40mgs until your symptoms improve considerably. You need to be 90 plus percent healed BEFORE you are off of pred entirely, otherwise your flare symptoms will return. And it sounds like you are not seeing improvements yet. So take that taper schedule and forget about it. Instead, watch your symptoms and listen to your body. Reduce pred slowly as you are healing and not until then.
Did exactly this. In the early days just one day of 40mg Pred would be enough to fix me right up. I didn't taper down after a day, obviously. Would be on 40mg for at least a week before I began the taper, but my symptoms still returned every single time.
Pred is such a troll drug.
Edit: And, no, I never split the dose. Never felt the need to.