Unless you live somewhere that sits on bedrock or whatever, I've heard some people plant daikon radishes for one season, then don't even harvest it and let it rot in the soil, and apparently, it's a good first step to turning compacted dirt into healthy soil.

I've also seen people plant corn in raised beds, so that's always an option.

Should be just enough. Plant at 2 inches deep and Just go wide (4-6ft) on the row & column spacing to allow them to spread out their roots horizontally.

Fold the corn stubble back into the soil at the end of the year to (slightly) increase soil depth for next year. Probably too shallow to do radishes as a cover crop for the first couple years. But you should be able to build things up eventually.

Do some raised bed stuff. Make some areas 4 inches deep. Or hugelkultur it.

https://en.wikipedia.org/wiki/H%C3%BCgelkultur

This Boomer HOA's...

2 inches of topsoil over packed clay, rock and sand.

so i have a friend who wrote a textbook on turning your yard into a garden you could live off of, if you've got about half an acre to an acre i think, it's been a while since i read the intro. you have to dig rows and such but it's doable. takes a few hours of work every day. dude grew the best watermelons