Who told you there is not enough protein in chick feed for hens? I suggest you look at the bags to see the analysis. I can't remember where you are located, but in the US they put labels on the feed that tells you percent protein among other things like calcium, salt, and fiber. There is usually a different section on the label that tells you ingredients, like corn, soy, or wheat. I'm talking about feed meant for chickens. There are some game bird feeds, meant for species other than chickens, that are different.
Layer normally has 16% protein. Chick Starter feed normally has over 20% protein. Chick Grower normally has 16% protein. There is one chick feed, a Grower/Developer that is sometimes available to feed them from 13 weeks until they start to lay, and it has 15% protein. That is what I use if I can get it for the entire flock once the chicks are old enough. But it is hard to find a chick feed that has less protein than Layer.
You do not want to feed Layer to growing chicks, especially on a regular basis. The only significant difference in Layer and Grower is that Layer usually has somewhere around 4% calcium and Grower probably has around 1%. That extra calcium can, over time, cause bone deformation or kidney problems in growing chicks. One or two bites is not going to harm them, but a steady diet can.
I feed chick Starter or Grower to the entire flock and offer oyster shell on the side. The hens that need the extra calciunm for their eggs will get that from the oyster shell and the growing chicks pretty much leave it alone.