Concentrating the cattle to one padock while resting the others. Longer grass builds more soil than shorter grass, basically the part the cows eat may be the same, but longer grass means more plant litter, more mature grass means more fiber which is the organic matter that goes to the soil. A lawn builds very little soil, the clippings have very little fibre in them, and the roots are short due to the balance between green parts and roots. In a pasture with long rests and short intensive grazing the grass builds massive organic matter both below and above ground.
Move the hay feeding around at on bale per spot with a ring of manure and prints around it, then the next such ring just touching the first. Manure works best evenly spread.
Legumes, yes, the more total production on the land, the faster the soil building.