IMO, plastic or so called solarization is the absolute worst thing one can do to garden soil. I would never ever do it to my garden soil. It kills everything, indiscriminately. Then weeds come back with a vengeance.
Fill that compost bin and keep it full. Turn it when you can. Add more and more. Look for sources of organic matter...it is all around you and usually free for the asking/taking.
Cover crops are great when you mow them and let the clippings go back into the soil. Then if ready, you can run the tiller over it or let it grow back again to mow again. Repeat, repeat, repeat. Mow it short when you know you are ready to till otherwise leave some to grow back Clovers, alfalfa, vetch, cereal grains, etc. are all great.
You could be growing peas right now and building that soil.... right now, today. When ready/mature mow them down with a lawn mower (don't harvest) and let them reseed to grow again and add tons of organic matter to your soil. I have done as many as 4 complete cycles of mow, let reseed, let regrow, mow, let reseed, let regrow, etc. . before frost without ever adding a single seed after the original planting...4 times...adding huge amounts of FREE organic matter to your soil each time. See below
Till them in when you want, Cow peas aka field peas such as blackeye, purple hull, crowder, etc. etc. They are all great soil builders, especially when you don't pick them but mow them and till them in or let them reseed. Make it a practice to
never ever leave your garden soil without something growing in it...either cover or produce.
Lime if you need ph adjustment but otherwise not. Peat moss is a waste of time and money. Get aggressive and get organic materials in there and keep adding them. Investing in your soil has great ROI, return on investment. It is a continuous process ever increasing the value of your soil.
Here is an example of mowed/shredded peas reseeding and adding valuable nutrient to your soil.
Here is what they look like after regrowing:
Anyone can do this...large plot or small. It doesn't require hardly any money just a few cents for seed but does require some time and effort. The rewards are soil tests which say, "No N-P-K required" and vegetables which are out of this world in taste and nutrition.