I think this is based on a false premise.
Without needing a degree in astrophysics, this simple explanation will do…
Pretty much by definition a new moon is at its zenith at noon & behind the earth at night.
To create a new moon, the sun has to be behind the moon from our perspective, so its visible face is in shadow to us. This means they have to be ‘up’ at the same time.
A full moon, by the same token is at zenith at night… right where you don’t want it if you need darkness.
A full moon is when the sun is shining right on its visible face, which means they have to be on opposite sides of the earth.
See TimeAndDate.com – London, England, United Kingdom — Moonrise, Moonset, and Moon Phases, June 2020 for an example.
So, if you wait for a new moon, it’s never going to be in the sky at night.
By this token you could probably get away with anything before the first quarter, or after the last quarter, though you will probably get some internal reflection in the atmosphere. To know how great an effect that will have on any given phase, you probably would have to ask an astrophysicist ;]