A few things:
1.) What do your levels look like before you go to bed? What I'm getting at is whether it makes sense to consider if your basal is too low.
2.) Do your morning spikes required correction, or is it temporary? Many people, myself included, have issues with dawn phenomenon and/or some sort of blood sugar spike upon waking. Mine is bad enough that I have to give myself insulin even though I haven't eaten in 8+ hrs.
3.) What does your typical day of eating look like and are you carb counting? Many people (again myself included) find that a strict diet has phenomenal positive effects on our blood sugar regulation and insulin management.