Final answer:
The moon phase after the new moon with increasing illumination and less than half visible is the waxing crescent phase. It leads up to the first quarter moon, where half of the moon's illuminated side is visible from Earth.
Step-by-step explanation:
The moon phase after the new moon when the sunlit portion is increasing and less than half is illuminated is known as the waxing crescent phase. This occurs after position A when the moon is in the same general direction as the Sun and is largely invisible from Earth because the illuminated side is facing away from us.
As the Moon moves along its orbit to position C, it becomes a first-quarter moon, where half of its illuminated side is visible to us. Between these two phases, during the waxing crescent phase, the sunlit portion visible from Earth is growing, but still less than half until it reaches the quarter phase.
The first quarter moon is characterized by its appearance in the sky around noon and setting around midnight. Following the first quarter phase, the moon enters the waxing gibbous phase, leading up to the full moon where it's completely illuminated and opposite the Sun in the sky.