Wrong basal or dawn phenomena or a night time hypo resulting in a morning rebound.
Can you get hold of a continuous glucose monitor, even if just for a week, so that you can tell what's happening at night?
Too low a basal will probably lead to a gradual rise over the night, dawn phenomena will be a more sudden rise, caused by the sugar released by the liver in order to help you get going in the morning. If your liver helps you out of a hypo at night, that can also cause a morning high.
Out of interest, had you had any alcohol in the nights with morning highs? Alcohol can often suppress the dawn phenomena, because the liver is too busy processing the alcohol to push out its morning encouragement. Of course, that also makes it ineffective at protection against night time hypos, so it's not necessarily a good idea.