I understand why even pastured/free range flocks need to be supplemented with grain during the winter - the bug population is pretty minimal in cold weather, so the chickens need some kind of supplementation.

Wonder if there might be a possibility during the summer of some younger chickens who never needed to be supplemented with grain? Some breeds can start laying eggs at 16-18 weeks of age, although at that age they'd be smaller chickens and produce smaller eggs.

Any chance you could raise a few of your own from chicks so that you can control their diets? (I live in an area that prohibits backyard chickens, but it might be worth looking into)