Between sunrise and sunset during the daytime, the Moon is rather hard to see with the naked eye if it is between first quarter and last quarter around the time of New Moon. The brightness of the Moon falls below the brightness of the sky during these times, so to see these phases you have to wait until just before sunrise, or just after sunset. In fact, there are international competitions among amateur astronomers to see with the naked eye the first trace of the lunar crescent. The earliest First Crescent I am aware of is the Moon about 17 hours after New Moon.