“Every situation is different, but for us, it was not an option to leave our pets behind,” Ms. Penn said. “They are my best friends.”

Other animals were not as fortunate.

A woman in Corpus Christi said on Twitter that she took in her neighbor’s dog left in the backyard. A photographer for The Daily Mail rescued a dog he found chained to a pole in Victoria, waters rising around him, the paper reported. In Dickinson, a CNN reporter spotted two retrievers abandoned in a boat.

In San Antonio, the city’s Animal Care Services Department had taken in about 200 displaced animals as of Monday afternoon. More were on the way.