This second moult is known as an eclipse moult and birdwatchers often refer to a bird as being in 'eclipse'. Most birds do not shed all their feathers at once. This would be dangerous. The exception to this rule is some female hornbills which spend time each year walled up inside a hole in a tree sitting on eggs. She is brought food by her mate and uses this opportunity to shed all their feathers and grow a complete new set.
Penguins also moult in a disorganised way.
They cannot afford to lose many feathers at one time because of the cold. They stand around in large groups with new feathers growing all over their bodies. These new feathers push out the old ones which the bird then removes by preening. Most birds moult by replacing a few feathers at a time. Moulting is normally symmetrical.