I agree, it is the green in their diet that adds to the flavor and color of your eggs.
Two things I do to help boost the "greens" in their diet at no added cost: keep a chicken treat bowl in your kitchen for all fruit and veggie trimmings, leftover cooked veggies, and any other food scraps they might like (which is just about anything). They happily eat all the trimmings, the seeds from squashes, bell peppers seeds and cores, apple peels/cores, radish leaves, strawberry tops, lettuce trimmings...you get the picture. Along with any other leftovers or slightly stale food stuff you may have. The only thing they haven't eaten is celery, I think it is just to hard for them. I take it out to them every morning and it's a race to see what goodies I have.
Look to your garden and yard for more greens, anytime I weed I dump my bucket into their run. Also let a little portion (no fertilizer or chemicals) of your lawn near your coop go, they'll happily eat as much grass, dandilions, clover and more as you can toss to them. Plant a patch of leaf lettuce, bok choy and so on, they'll eat this tender greens with great gusto.