I did, years ago, plant Winter Rye as a winter cover crop (aka green manure) and tilled it in in the spring and I know many others that still do. You can till in the spring when the soil is workable, but that also needs to be several weeks before you plant.

I would think that alfalfa might not make much mass by fall.

I will plant some daikons [tillage radishes] soon. They make some top bulk, but especially makes a deep root which makes a good bulk in the soil and they winter kill [in the north] and do not delay spring planting.

Well, with all the hot weather and then the rain that we got, the alfalfa is beginning to sprout. If it grows as quickly as it sprouts, I should have a nice biomass to till in come fall.

I have been doing the planting of alfalfa and buckwheat for the last two years, and I will plant late in August, so that the plantings are well established before the frost kills it off. And about March, I turn it over in each row. The soil is pretty workable at that time, and I only use my hoe on my 100' x 35' garden. I should add that in the Spring, I have been putting in about 20' of buckwheat, which I will let grow to seed. I cut off the tops of the plants when the seeds are ready. Then I'm providing enough seed for the Fall plantings. I have liked the results.