This is only your first morning test! Don't be scared.
Our fasting readings are always the last to behave because they are determined by lots of factors over which we have no control. Restless night, bad dream, testing immediately you get up after washing hands thoroughly (important), and most of all the Dawn Phenomenon. This is when our livers dump glucose in our bodies to help us start the day after a fast. Some livers produce more than others. It can also happen after exercise and at other times when our livers think we are a bit short of glucose. It is perfectly natural, it happens to non-diabetics too. It may happen more often at the beginning because our bodies are used to higher levels of glucose, so when we start a low carb diet and reduce our levels it takes a while for our livers to catch up.
One way to see if it is your liver is to test just before you go to bed. See how much it goes up over night,
6.5 is not bad. Many on here are much higher. Have patience, it will come down if you stick at it, and stressing about it will make it worse.