Up until about 5 years ago, we did conventional ag (plowing, discing, sowing and cultipacking). For the last 5 years, we have been half-way doing regenerative ag (spraying, sowing and mowing - and occasionally lightly disc to expose some soil). We have had a good mulch layer in the last 5 years that helps retain soil moisture and protects against soil wind and water erosion. The ground does seem quite softer, which typically indicates healthier soil, as opposed to it being rocky and hard (that it has been prior to the last 5 years. We took the plunge and bought a no-till drill last year and started no-tilling seed into the fields. Simply spray and drill. Our next purchase will be a roller-crimper, which we hope to have before this coming spring. That will complete the process of no applicants or additives. Just planting different forages for different root systems. Look up growingdeer.tv and check out his food plot videos - they are very informative!
Rye will definitely help build the soil - it will produce a TON of biomass in the spring that will add a lot of mulch and carbon to the ground/soil. That and some crimson clover, which adds nitrogen to the soil. Both of those will grow anywhere and put on a lot of biomass. You'll need to terminate the rye in the spring after it gets high and before seed is formed. Then just plant some buckwheat, or something that is fairly drought tolerant.
Come spring, we will drill our soybeans into our fall mixture that is 4-6 ft tall. A week or so after, we will crimp it (terminate it). The rye acts not only a weed suppressor (3-5 inches thick), but it also releases allelopathic chemicals which prevents small weed seeds from germinating. Once the soybeans are grown, come late summer or early fall, we will drill our fall crop right into the soybeans.
Essentially, you always want to have a living root, which helps the soil, but also you will have something for the critters to eat year-round.
