Sunday June 3rd the waning gibbous Moon will be crossing the plane of the ecliptic moving south. This is known as the descending node, one of two intersections the Moon’s orbital path (dark green line) has with the ecliptic.
On the morning of the node crossing the 19-day old waning gibbous Moon will be over the south-southeastern horizon about an hour before the Sun rises local time. The Moon will also be about 3o from the ‘red planet’ Mars. Off to the west is the planet Saturn nestled within the glow of the eastern side of the Milky Way.
Depending on how dark the sky is where you are viewing from and how early you want to go outside, the area around Saturn is rich with some of the best deep-sky objects visible with binoculars. So the earlier you are out, before moonrise, the darker the sky will appear making it easier to see some of Messier Objects in the area near Saturn.