(Updates with shelter details, increase in power outages)
* Irma downgraded to tropical storm as it heads northwest
* Nearly 6.5 million in Florida, other states without power
* 38 killed in Caribbean, one in U.S.
FLORIDA CITY/ESTERO, Fla., Sept 11 (Reuters) - Displaced Florida residents started heading back to their homes on Monday as a weakened Hurricane Irma advanced inland, flooding several cities in the northeast of the state while millions of people remained without power.
Downgraded to a tropical storm early on Monday, Irma had ranked as one of the most powerful Atlantic hurricanes recorded. It cut power to millions of people and ripped roofs off homes as it hit a wide swath of Florida on Sunday and Monday.
Authorities said the storm had killed 38 people in the Caribbean and one in Florida, a man found dead in a pickup truck that had crashed into a tree in high winds on the Florida Keys over the weekend.
With sustained winds of up to 60 mph (100 kph), Irma had crossed into Georgia and was situated about 47 miles (76 km)northeast of the Florida state capital Tallahassee, the National Hurricane Center said at 11 a.m. ET (1500 GMT).
High winds snapped power lines and left almost 6.5 million homes and businesses without power in Florida, Georgia, South Carolina and Alabama, state officials and utilities said. They said it could take weeks to complete repairs.
Miami International Airport, one of the busiest in the country, halted passenger flights through at least Monday.
Police in Miami-Dade County said they had made 29 arrests for looting and burglary. Fort Lauderdale police said they had arrested 19 people for looting.
About two dozen vehicles filled with people hoping to return home after fleeing the Florida Keys, where Irma roared ashore on Sunday with sustained winds of up to 130 mph (209 kph), lined up near the entrance to the highway that connects the archipelago to the mainland with a series of bridges and causeways.
They expressed anger at police who asked them to drive to a racetrack a few miles away to register first.
"This is how people are going to die - nobody's going to want to leave the Keys," shouted Shelby Bentley. "I've been in the Keys for 40 years ... It's the first time I've evacuated from a hurricane. It'll be my last time."
Officials in Monroe County, where the Keys are located, said most of the islands still lacked fuel, electricity, running water and cell service on Monday.
"Supplies are running low and anxiety is running high," the county said in a statement posted online.
Irma hit Florida after powering through the Caribbean as a rare Category 5 hurricane. It killed 38 people, including 10 in Cuba, which was battered over the weekend by ferocious winds and 36-foot (11-meter) waves.
A week earlier Hurricane Harvey flooded a wide swath of Houston. Nearly three months remain in the official Atlantic hurricane season.
Northeastern Florida cities including Jacksonville were flooding on Monday, with city sheriffs pulling residents from waist-deep water.
"Stay inside. Go up. Not out," Jacksonville's website warned residents. "There is flooding throughout the city."
BILLIONS IN DAMAGE
The storm did some $20 billion to $40 billion in damage to insured property as it tore through Florida, catastrophe modeling firm AIR Worldwide estimated.
That estimate, lower than earlier forecasts of up to $50 billion in insured losses, helped spur a relief rally on Wall Street as fears eased that Irma would cut into U.S. economic growth.
Shares of insurance companies were among the big winners, with Florida-based Federated National, HCI Group and Universal Insurance all up more than 12 percent.
Some 6.5 million people, about one-third of Florida's population, had been ordered to evacuate their homes ahead of Irma's arrival. More than 200,000 people sought refuge in about 700 shelters, according to state data.
As shelters began to empty on Monday, some 7,000 people filed out of Germain Arena in Estero, south of Fort Myers. The crowd included Don Sciarretta, who rode out the storm with his 90-year-old friend, Elsie Johnston, who suffers from Alzheimers disease.
Sciarretta, 73, spent two days without sleep, holding up a slumped-over Johnston and making sure she did not fall out of her chair. He relied on other people in the shelter to bring the pair food, often after waiting on hours-long lines.
"For the next storm, I'll go somewhere on my own like a hotel or a friend's house," Sciarretta said. "I'm not going through this again."
Shelters across western Florida opened, filled up and often closed because of overcrowding after the storm made a western shift on Saturday. The jam-packed Germain Arena, a sea of cots and blankets, closed to new occupants on Saturday night.
U.S. President Donald Trump, attending a ceremony at the Pentagon on the 16th anniversary of the Sept. 11 attacks, vowed a full response to Irma as well as ongoing federal support for victims of Hurricane Harvey, which flooded Texas.
"These are storms of catastrophic severity and we are marshaling the full resources of the federal government to help our fellow Americans," Trump said.
Florida's largest city, Miami, was spared the brunt of the storm but still suffered heavy flooding. Boats, including one named "Lucky Duck," had been driven ashore by the storm and utility crews were on the streets there clearing downed trees and power lines.
(Additional reporting by Daniel Trotta in Orlando, Bernie Woodall, Ben Gruber and Zachary Fagenson in Miami, Letitia Stein in Detroit, Colleen Jenkins in Winston-Salem, N.C., Doina Chiacu and Jeff Mason in Washington, Scott DiSavino in New York and Marc Frank in Havana; Writing by Scott Malone; Editing by Frances Kerry, Paul Simao and Howard Goller)