That situation is going to be a long term maintenance challenge. With little soil to root into the St. Augustine will require lots of watering to keep it alive and healthy. I might consider pulling up the sod with a sod cutter and setting it aside where you can keep it moist for a couple of days. You might also want to remove some of the larger surface rocks and debris before adding the extra soil. Then bring in more soil to spread over the area to a depth of at least 6 inches (more is better). Then lay the sod back down and water regularly for a few weeks to get it reestablished. There are usually some local places to rent a sod cutter to make pulling up the lawn a little less work.
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