Help with dead areas after heat/little water

Howdy everyone. I have my back yard with larges portions of dead grass after that brutal stretch of July/August of heat and little water.

Location: New Orleans area
Grass type: St. Aug (with some Bermuda I think creeping in)

I am curious as to the best/quickest/healthiest way to begin getting the grass to where it was.

Should I spread some dirt/sand/mulch on the dead areas and water regularly? Get a mulcher, flatten it out and add new sod?

Thanks in advance for any help/advice.