Mine have been worryingly high in the morning, I am on four injections a day and take lanctus every 24 hours as a background insulin. My bg returns back to normal in the mornings if I put I change the dosage of the lanctus.
I think it will l depend on what insulin your taking and the dosage, maybe put the dosage up and see if it solves the problem or you may be having hypos in the night and bouncing back up from that?