Tarrant County officials said last month that a programming glitch caused some ballots to be counted as many as six times. The county's vendor for voting machines took the blame.
After the results were corrected, Justice Willett's statewide margin of victory shrunk from 21,000. But Mr. Smith says he took more than 38 percent of the vote in Tarrant County.
"In a race this close, we think it's more important than ever that every single vote be counted correctly," said David Rogers, Mr. Smith's campaign manager.
Mr. Rogers said the lawsuit has a "1 in 5 chance" of succeeding. He said more results would have to change in Tarrant County for Mr. Smith to win.
Jeff Fisher, executive director of the Texas Republican Party, said he did not believe there were any more problems in Tarrant County. He said Mr. Smith's lawsuit would not change the election's results.
Mr. Smith also seeks to inspect election records in Gregg County, where Justice Willett more than doubled Mr. Smith's vote total. Mr. Smith argued he had done much better in Gregg County in past elections.
Mr. Smith, who served one term on the court after being elected in 2002, has 10 days to gather evidence of mistakes or fraud.
Meanwhile, State Rep. Carter Casteel has officially lost her primary for HD73.
The recount slightly increased the margin of victory of Nathan Macias over Rep. Carter Casteel. Mr. Macias' campaign received heavy financial support from James Leininger, a school voucher advocate from San Antonio. Ms. Casteel was a leader of a successful effort to defeat a voucher bill last year.
Ms. Casteel complained that the money allowed Mr. Macias to run attack ads against her.
Ms. Casteel said mailings paid for and coordinated by Mr. Leininger distorted or lied about her voting record, including saying she had voted against a parental consent requirement for minors to receive abortions. She voted for the bill.
"If I knew what I knew today, would I change what I did? Absolutely not," she said. "I would have stood up to Leininger like I did at the end of May."
She added: "If we don't moderate our party to some degree, we are going to be in so much trouble. Well, I say trouble. We're just not going to be in power."