It is well known that the most efficient way to pack circles in the plane without overlap is to arrange the circles so that their centers form a regular hexagonal lattice with spacing $2r$, where $r$ is the radius of the circles. Then fraction of the plane covered by the circles is $\frac{\pi \sqrt{3}}{6} \approx 0.907$. My question is related. Suppose we must cover the entire plane with no gaps, but overlapping of the circles is allowed. Then we may define the efficiency $\eta$ of such a packing to be the ratio
$$ \eta = \frac{\text{Area covered}}{\text{Area of the circles}} $$
Then we must reduce the spacing of the hexagonal lattice to $r\sqrt{3}$ so that the circles can cover the gaps. Now any given hexagon in the lattice with area $ \frac{3\sqrt{3}}{2} r^2 $is covered by a circle of area $\pi r^2 $. So the efficiency of this packing is
$$ \eta = \frac {(3\sqrt{3}/2) \cdot r^2}{\pi \cdot r^2} = \frac{3\sqrt{3}}{2\pi} \approx 0.827$$
Is there any other way to arrange circles to cover the entire plane with a greater efficiency, or is this the best possible? Searching the topic brings up papers that are above my level of understanding, or which focus on packing without overlaps.