What is the best way to correct the soil around a tree that's been "volcano mulched"?
See this image, courtesy of Madison Tree Care & Landscaping, for bad and good practice in tree mulching.
You should leave only a thin layer of mulch at the base of the tree, and avoid having the mulch directly against the trunk.
One rule of thumb is to mulch your tree using the 3-3-3 technique: no more than 3 inches thick, 3 feet wide, and at least 3 inches from the base of the tree.
See links below for our subject guide on Trees as well as two good articles on how to properly mulch around a tree.
Scott Aker of the Washington Post recommends this manner of mulching a tree: The correct method is to form a circular ridge of soil at the edge of the root zone and then lightly mulch this saucer so that the soil and mulch together trap rainwater and feed it to the roots. Mulch should not touch the trunk. I hope this information will help you revive and save the tree.
To learn more about proper tree care, see the "Tree Tenders" link below, and take our 9-hour course on this topic.