The next time you awaken feeling sick, just think through what your activity levels were for the whole of the preceding day (24 hours). Was it more than usual?
If this seems to be the case, why not do a night time b/m at 3am or 4am after especially active days to see if you are going low? After a few trials, you might even find when in the night this is happening - although I'm afraid we haven't pinned it down exactly - except that I think it is after 4am for my son.
Even if you don't have the flexibility of a pump, you could then perhaps have a bedtime snack (or bigger bedtime snack than normal) or slower-acting carbs etc, to try and avoid the hypo. Or maybe you could reduce your long acting insulin at bedtime?
I do know that my own findings are very much circumstantial - i.e. I have drawn my own conclusion about what is happening to my son. Nevertheless, I feel I must've found the answer for us because he was having days off school for the sickness quite regularly and now it is increasingly rare. I ask him now which type of sport he has done and we turn down the pump by either -10% or -20% for the whole of the rest of the day and night after the activity. Sometimes, this has not been entirely necessary and he's gone a bit high - but much better to do that once in a while than to have night-time hypos, me-thinks.