Let the radius of the circular piece of cardboard be $r$. The diagram shows a sector of the circle which would make one hat, with the minor arc shown becoming the circumference of the base of the hat.
The circumference of the circle is $2\pi r$. Since $$6r<2\pi r<7r$$ we can cut out $6$ hats in this fashion.
Moreover, the area of cardboard unused in cutting out any $6$ hats is less than the area of a single hat. Hence there is no possibility that more than $6$ hats could be cut out.