The Hungry Tide by Amitav Ghosh — Not set in Madurai but in the Sundarbans, it is still the best contemporary novelist's primer for travel in south and eastern India: rivers, religion, language, the gap between visitor and place.
Silappadikaram — One of the great Tamil epics, written by the Chera prince-turned-monk Ilango Adigal probably in the 5th century. The story of Kannagi, who tears off her own breast to burn the city of Madurai in revenge for her husband's murder, is the foundational Madurai myth and explains why you will see her statue on the Vaigai bridge.
A Hindu View of the World by S. Radhakrishnan — Not specifically about Madurai, but the philosopher-president's clear introduction to Hindu thought is a useful frame for anyone trying to make sense of what they see in the Meenakshi Temple.
The Madurai Sultanate: A Concise History by R.D. Choudhury — A short academic survey, but the only readable account of the short-lived 14th-century Muslim kingdom of Madurai, an episode often skipped in temple-focused histories.