At least 33 people were killed when Cyclone Thane hit north coastal Tamil Nadu and neighbouring Union Territory of Puducherry on Friday.
Heavy rain and wind gusts up to 140 km an hour hit the coast between Cuddalore, about 170 km from Chennai on the east coast, and Puducherry between 6.30 and 7.30am.
The cyclone also disrupted rail and air operations, with services cancelled or rescheduled.
"There could be some more deaths," said Marri Shashidhar Reddy, vice-chairman of the National Disaster Management Authority, which has sent eight teams to the coastal districts.
Cuddalore bore the brunt of the cyclone, accounting for 21 deaths, while Puducherry recorded seven deaths, Villupuram and Tiruvallur (two each) and Chennai (one) in mainly wall collapse and electrocution, said officials.
The storm, however, left Andhra Pradesh unscathed, which along with Tamil Nadu was on high alert for the last two days.
Announcing a relief of R2 lakh to the kin of victims and Rs 150 crore for immediate relief work, Tamil Nadu chief minister Jayalalithaa said 26 people had been killed due to the cyclone in four districts.
Puducherry was cut off from neighbouring districts in Tamil Nadu as uprooted trees blocked road traffic.
Puducherry chief minister N Rangasamy also announced a relief of Rs 2 lakh to the families of the deceased.
In Chennai, a cargo ship ran aground off Marina Beach with the sea conditions remaining rough.
After the landfall, Thane moved further westward and weakened into a deep depression, the regional weather office in Chennai said.
"The system is likely to weaken further," it said, forecasting heavy rainfall for the next 12-24 hours in different parts of Tamil Nadu and Puducherry.
(With agency inputs)