Flood waters looked ready to recede on Saturday, but forecasters said that with a cold front due to move into the area there could be more rain in store.
Many were set to spend another night on their rooftops, with tens of thousands already crammed into emergency shelters struggling to provide enough hot meals and dry beds.
One group stranded on a roof held a banner reading: "Enough. There are children, pregnant women, sick women. Send the police."
Health officials have voiced concerns about looming health risks from open sewage and the spread of disease-carrying mosquitoes.
|About 7,500 military and police are working|
on rescue and security operations [EPA]
Dengue, cholera and diarrhoea outbreaks are now real possibilities, they said.
Jose Angel Cordova Villalobos, the health minister, admitted that shelters were jammed and said authorities were planning to open new emergency facilities at sports arenas and possibly at a bull ring.
During the crisis, about 69,000 people have managed to get into about 600 government shelters, Beatriz Zavala, the federal social development minister, said.
Many fled flooding in Villahermosa, the state capital, for less affected areas.
Mauricio Hernandez, one resident fleeing the city, said: "We are leaving because we cannot live like this. We don't have any water, and the shelters are full."
Thousands of people have fled Tabasco into the neighbouring states of Chiapas and Veracruz seeking refuge.
But many others remained despite the flooding, refusing to leave for fear that looters would take all their possessions.
Around 1,000 people reportedly overran a shopping centre in Tabasco, overwhelming police posted in the area, and stealing items including televisions.
Smaller looting incidents occurred Friday and police made several arrests, the governor also said.
Felipe Calderon, Mexico's president, said on Friday there were 7,500 military and police staff working on rescue and security operations in Tabasco.