Beto O’Rourke may be just another private citizen now, after unsuccessful bids for president in 2020 and the U.S. Senate in 2018. But his response to the grid failure in Texas is rekindling speculation about his political plans.

While Republican leaders in Texas have come under fire this week over their handling to the crisis, Mr. O’Rourke, a Democrat from El Paso who served in Congress from 2013 to 2019, organized volunteers to make more than 784,000 wellness calls to senior citizens around the state.

Mr. O’Rourke also took to the airwaves, lambasting Senator Ted Cruz, his Republican opponent in the 2018 Senate race, after Mr. Cruz slipped away to Cancún while millions of Texans endured blackouts and water shortages.

In an oil-rich state that Republican leaders often extol as a cutting-edge energy colossus, the sense of alarm around the crisis — especially in major Democratic-led cities in Texas — is giving Mr. O’Rourke a chance to go after opponents on the right.

“The energy capital of North America cannot provide enough energy to warm and power people’s homes,” Mr. O’Rourke said on MSNBC. “We are nearing a failed state in Texas. And it has nothing to do with God or natural disasters. It has everything to do with those in positions of public trust who have failed us.”

He also publicly thanked Representative Alexandria Ocasio-Cortez, a Democrat from New York, for helping to raise $2 million for Texas relief efforts.

Mr. O’Rourke, who came within three percentage points of beating Mr. Cruz in 2018, has left open the possibility of mounting a challenge to unseat Gov. Greg Abbott, a Republican, in 2022.

Mr. O’Rourke has faced skepticism over his chances in a state where many people take pride in owning guns. He has faced criticism for urging greater control of assault-style weapons after the 2019 massacre of 23 people at a Walmart in El Paso. “Hell, yes, we’re going to take your AR-15, your AK-47,” he said in a presidential primary debate in September 2019.

But Mr. O’Rourke has also hit back at critics, including Mr. Abbott. In one barb on Twitter aimed at the governor in January, Mr. O’Rourke said, “You are obsessed with pleasing the NRA and the gun lobby instead of protecting the people you were elected to serve.”