I can only give you my experience - I'd guess that it's not the time you set the alarm for that causes it. First year after diagnosis my FBG was around the level you describe - I couldn't see any pattern to it. BG back in non-diabetic range didn't affect things. I had the dawn phenomenon glucose dump - it woke me up anytime from 3.30am onwards. Daylight seems to have soemthing to do with that.
In the last six months the FBG has dropped back to 4.8-5,6 territory - still get the dawn phenom, though. It does seem to be something that takes most time to drop back.
Does for me, as I never seemed to have rid myself of Dawn Phenomenon. My liver starts dumping until I eat, whether it's 2 a.m. or 6 a.m., which can be a bit of a pain. That explained my HbA1c of 35 when I usually see 4's and 5's during the day. Turns out I have night-time sixes. So yeah, higher blood sugars can happen.