I think you need to talk to your medical team about this. If you can't link the results you're seeing to (say) food intake or medication then there is something else in play. No idea what that might be. The variation you report, which - if I've understood you correctly - can be 3.6 at midnight to 15 or 16 in the next few hours is quite large, particularly if you've not eaten anything or taken any medication that might help explain it.
Incidentally, I think you need to focus on your actual high and low readings, rather than an "average" which can present a false picture. For example, four readings at 5.0mmol/l and four readings at 15.0 average to 10.0mmol/l - but none of the actual results is 10. The HbA1c isn't an average of anything, by counting glycated red blood cells it gives you a longer term idea of what your blood glucose has been doing over the last three months, with a big skew towards the most recent few weeks.