The first quarter moon rises around noon. Roughly half of all the Moon's time above the horizon is spent in a blue sky, and almost nobody looks up.

Only the full moon keeps strictly to the night, rising at sunset and setting at sunrise. Every other phase shares the sky with the Sun for part of its run. A first quarter moon is due south at sunset, which means it has already been up for six hours. Once you know to look, you start seeing it constantly, and it is faintly annoying that you didn't before.