Veda Rose Knappenberger lost everything in March when a tornado tore up her house in Cave City, Ark., leaving her bruised and shaken to the core.

A neighbor, Kathy McLeod, invited Ms. Knappenberger, 78, to sleep on her couch until help arrived. But by then, the Federal Emergency Management Agency had delivered startling news: It was denying assistance to residents of the nine counties hit by the storm system, saying the damage appeared contained enough for state and local officials and volunteers to handle.

Gov. Sarah Huckabee Sanders, a Republican who served as President Trump’s press secretary in his first term, appealed the denial. Another month passed. Governor Sanders made a personal appeal in a phone call with her former boss. Shortly after, on May 13, Mr. Trump announced he had approved a disaster declaration for Arkansas, allowing residents to apply for a type of FEMA aid known as individual assistance.

“They shouldn’t have denied it — they should have at least said, ‘We’re working on it,’” Ms. McLeod said upon learning that federal help was coming after all. “That way, people wouldn’t have thought they were just forgotten.” She would tell Ms. Knappenberger the news when she picked her up from physical therapy, which she needed in part for injuries she suffered during the storm.