Answer: As you have discovered, shade lovers will not tolerate that brutally hot afternoon sun, even for a few hours. So your selection needs to be sunlovers that will tolerate a bit of shade since they are not getting a full days' sun -- just the hottest part.
The plants are probably also having to compete with the tree's roots for water and nutrients and the tree canopy probably stops a lot of the rain from reaching the ground so the area is a bit dry. So you will need to select plants that are very vigorous and can try to outcompete the tree -- such as the phlox.
Whatever you plant, you will need to prepare the soil carefully, working in a generous amount of organic matter and then also use several inches of organic mulch. The mulch will help to conserve soil moisture and also provide an ongoing source of organic matter as it breaks down. While the plants are becoming established you will need to water them to keep the soil evenly moist but never soggy. Use your finger to check under the mulch and into the soil to see if and when you need to water. Watering deeply less often is better than a daily sprinkling so check to see how deep your watering actually soaks in, too.
Here are some perennial plants to experiment with: achillea or yarrow, perennial geraniums, daylilies (hemerocallis), black eyed Susan (rudbeckia) and purple coneflower (echinacea purpurea), and amsonia. Perennials each have a relatively short season of bloom, but they are generally more tolerant of adverse conditions than most annuals. Annuals to try might include a verbena such as "Homestead Purple", salvia (many types), and marigolds. The annuals will require regular watering and feeding and deadheading to bloom their best.
Another possibility would be to use a groundcover under the tree, perhaps more of the same creeping phlox that you already know will survive there, and then set an attractive container nearby for the summer color. The container could be located so as to receive more consistent light during the day and then you would have a better chance of selecting annuals that would thrive and bloom nicely all summer. You could even set a bench under the tree where you could sit to admire the scenery -- and add all season interest to the area.
I hope this gives you some ideas.
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