“It wasn’t like I thought, ‘Hey I’m just going to party in Florida and forget this vote,’” he said.

The trip to Florida, which included an awards ceremony, dinner and visit to the Disney World complex, was a reward from Mr. Farmer’s employer, Barton’s Lumber, a chain with stores in Arkansas and Missouri, for his being named the best assistant store manager in his district, he said.

Despite his inability to cast his own ballot, the importance of each and every vote hadn’t been lost on Mr. Farmer, who urged his wife, Sara Farmer, to make sure her voice was heard.

“He told me, ‘Make sure you vote — if I lose by one vote, it’s going to be on you,’” Ms. Farmer, who voted early herself, said in an interview Wednesday.

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Hoxie, in northeast Arkansas, is home to about 2,700 people and is about a 90-minute drive from Memphis, where the Farmer family’s flight landed on Election Day last week. Its City Council has six members who handle the kinds of issues that most similar bodies face, such as passing ordinances, fixing roads and determining how city funds are allocated.

In such small towns, campaigning is often a low-key affair. (The story of the runoff’s results and Mr. Farmer’s missed opportunity was picked up by a local newspaper, The Jonesboro Sun, and some national news media.)

“You pretty well know everybody,” Ms. Linebaugh, a two-term council member, said in an interview on Wednesday. “It’s mainly just about who knows you and if they think you’ll do a good job or not.”