I would not move her.
But it wouldn't hurt to give her some starter.
Many of us feed a 18-20% all flock type feed,
with oyster shell in a separate feeder for the active layers.
Chickens molt and grow a new set of feathers 2-3 times before around 6 months of age, those are called the juvenile molts.
Some pullets will have a partial molt their first fall/winter.
Most birds will have their first full adult molt their second fall/winter.