You could add a good quality compost and it might "get the lawn going a little quicker", but that depends on the Soil Food Web, those wee critters that convert organic matter into plant food. It takes time for the SFW to decrease, as the level of organic matter in the soil decreases, as they digest that OM. It also takes time to rebuild that SFW once it is depeleted and that takes whatever time Ma Nature determines. There is not much we can do to hurry that process.

thank you for the reply, all the great information is making a little more sense.

I replied to your other message. Again, relax. Just fertilize, water, and mow for now. Even a lawn full of weeds looks much better when you fertilize, water and mow. Add grass seed in the late summer.