When splitting the background insulin, it's normal to find that both 12 hour periods aren't identical. It's not easy to get the daytime period correct as they're are so many variables to consider: food eaten, activity levels, strenuous exercise, stress etc. It's easier to work out how much to take before bed, but even then - this can still be a challenge with dawn phenomenon, night time lows, etc.
It would sound like upping the morning dose might help, but personally; I would be checking in with the diabetic team before making any changes. There might be something obvious here to a medical professional.
Grant