There is an organic (?) solution to compacted soil. I have forgotten what it is, but you basically spread it over the soil in compacted areas and it helps to 'break up' or aerate the soil.
Compacted Soil-What to Use (Organic)
Certified GKH Gardening Expert
This article will help: https://www.gardeningknowhow.com/garden-how-to/soil-fertilizers/improving-compacted-soil.htm
Gypsum i believe, but there is debate about tilling to incorporate compost
What I do is mix compost vermiculite,compost, and gypsum and them incorporate everything, I also add osmocote?? lol you know the fertilizer that is little balls and wont burn your plants. I guess it depends how compacted is your soil. My backyard is black clay, gumbo and god only knows what else.
You can also (this is for next season it will take a year to compost but is worth it) Put down all the leaves you collect and grass clippings next goes all the card board you can get your hands on, news paper(no glossy stuff) last cover it will a thick layer of black plastic. Mine is going right now... And next season you uncover it and it will nice to plant in the spot! weeds free and good soil!!