Maybe I am stating the obvious, but it is important to remember your levels go up and down all the time. You do not have to eat for them to rise.
Even in a healthy body there is no single correct level at which the body can keep your levels. They go low so glucose is released, they go high then insulin is released. Up and down and up and down. That is why normal is regarded as anywhere between 4 and 7 mmol/L. Because some naturally people just end to be lower, other people tend to be higher, and no one stays the same.
If your levels just kept going lower over night you would become hypoglycaemic and risk waking up in a coma. Albeit without waking up. So your body dumping glucose into your system is a good thing, it is keeping you alive. It is only a problem if you wake up too high, which 6.5 mmol/L is not. It is around the upper end of normal, but it is still normal.
Do not take targets literally but as general aims. You physically cannot control the fractions, and as Kalyz says, all three of your readings fall within the same margin of error for a glucometer. Your actual levels on all those occasions could actually have been 6.0 mmol/L. But you will never know for sure, and no matter what you do your body will be releasing glucose and insulin as it deems necessary, going up and down and up and down. Nothing you can do can make it stop on 6.0 or any other number.
To me it sounds like you are achieving exactly what your G.P. wants yo see.