Well, Fiona,
I'll try to tell you what I know (not saying that that's much, lol :p). It sounds to me like you might be going low in the middle of the night. To compensate for that your liver releases sugar it has stored for such occasions. Needless to say , many people rebound from being low to the other extreme. Have you tried setting your clock for 2 or 3 am and testing? You may be surprised to find you are low at that time.
I know that's a pretty basic explanation, but I hope it clears things up a little. I know others here can give you a better, more detailed explanation of this, so hopefully you'll hear from them soon.
Take care.
