WASHINGTON — David McCormick had a clear explanation for why his fellow Republican, Mehmet Oz, lost a critical Pennsylvania Senate seat: Voters viewed the daytime television celebrity as an interloper from New Jersey with limited ties to the state he hoped to represent.

“People want to know that the person that they’re voting for ‘gets it,’” Mr. McCormick, who narrowly lost to Mr. Oz in a GOP primary, said in March when asked to offer a postmortem of the general election defeat. “And part of ‘getting it’ is understanding that you just didn’t come in yesterday.”

As Republicans aim to gain the one seat they need to retake the Senate in next year's elections, Mr. McCormick is a top recruit. And before his anticipated campaign, he's working to avoid Mr. Oz's fate, frequently noting his upbringing in Pennsylvania, his ownership of a home in Pittsburgh and a family farm near Bloomsburg.

“I live in Pennsylvania,” Mr. McCormick said during a March appearance on Texas Sen. Ted Cruz’s podcast. But the reality is more complex. While Mr. McCormick does own a home in Pittsburgh, a review of public records, real estate listings and footage from recent interviews indicates he still lives on Connecticut's “Gold Coast,” one of the densest concentrations of wealth in America. The former hedge fund CEO rents a $16 million mansion in Westport that features a 1,500-bottle wine cellar, an elevator and a “private waterfront resort” overlooking Long Island Sound.