If you are constantly having hypo's the following morning, then this is a time delay on your exercise...
You need to look at what changes you might be able to do, but this will depend where what your bg's are like and when the excercise delay kicks in.. It might be possible to reduce your insulin to a lower ratio with your evening meal, but you would need to check that you aren't going to high before the delay kicks in...
Another option is to have a snack before bedtime depending on your BG this could be carbs, or a snack higher in protien which will adsorb slower might do the trick..
It's worth checking your BG, before betime, 2am-3am (this is around the time everything is at it's lowest ebb) if necessary check at other hours of the night as you might be able to identify at what point the delay start taking effect... and give yourself a better idea to what you need which is your best option