Do you take your Lantus dose once per day? It's well known that almost all basal insulins don't last 24 hours; with the exception of Tresiba I believe. Seeing as your BG raises around midnight, this could well coincide with the prior 10pm basal dose running out whilst the latest one waits to kick in.
As rightly suggested by
@azure, you should check in with your DSN and they may put forward the idea of splitting your Lantus dose in half to provide better coverage.
Do let us know how you get on