Next time you're in the feed store, read the ingredients labels on chick starter and Purina Flock Raiser (an all flock feed). They are practically identical. In fact, the Flock Raiser has more protein. There have been times when the feed store was out of the all flock feed and I bought chick starter to feed to my flock.
If you normally feed layer feed to your hens, you would then want to buy chick starter to feed any new chicks since the high calcium in layer isn't good for growing chicks. So, it would be entirely acceptable to then feed chick starter to both chicks and laying hens until the chicks start to lay. At that point, you could go back to feeding layer feed, or continue to feed chick starter with a side of oyster shell so the layers get the calcium they need in order to build eggs.