You may want to read my post on tillage radishes, as it ties in with what I am talking about here.

In a polyculture with onions, garlic, shallots, chives, rhubarb and peas (as the winter active crops). Right now I have raspberries in the system they don't compete very much with the winter active crops but they also don't pay much. Jerusalem artichokes don't affect the yield much either, but they don't sell that well and you can only eat so much. Also neither one die off in August giving way to the winter active crops which start growing in August or September.

Corn on the cob can be cut down in August or before
Beans often finish by that time
Zucchini, are mostly finished by then (water)
Leafy greens, alot of them could fit the time line but they need water (we often have droughts here from June until September).
Tomatillo, I just came up with this one they usually die off promptly after producing one big flush of fruits
Um... drought tolerant... dies/finishes in August or September
Okra, eureka!!! okra is drought tolerant, it has a tape root. It will die off a bit late but I think this Is the best idea yet for a short warm season crop while the primary winter active crops are dormant.

Please give suggestions if you can think of anything preferably with a big root system and weedy/easy to grow.

Potatoes! You can plant them in deep mulch with more added as they grow (I've used straw before). By the time they're done you've got lovely rich soil with tons of microbial life.

In my gardens, the summer squash would be done by the end of August - I could never keep the plants more than a few weeks after they started producing - too many marmorated stink bugs carrying diseases.