Yes, there's a name for that transitional phase when most of us think the moon is full but it's not quite there yet. Unless the moon is 100 percent lit, it's in the gibbous phase. When the illuminated part of the moon is growing larger, approaching the full moon, the phase is known as "waxing gibbous." It appears high in the east at sunset, one to two weeks after the new moon.
When the illuminated part is shrinking, approaching the new moon once again, this phase is known as "waning gibbous." You'll see it in the eastern horizon after sunset, a few days after the full moon.