In an interview with The Courier-Journal of Louisville, Ky., on Wednesday, Ms. McGrath offered a glimpse at how she may navigate running as a Democrat in a red state. She criticized President Trump on foreign policy, but declined to label herself “anti-Trump,” and she said she did not support “Medicare for all” or offering subsidized health insurance to undocumented immigrants.

She also said she would have voted to confirm Justice Brett M. Kavanaugh to the Supreme Court, saying she saw “nothing in his record that I think would disqualify him in any way.” The comment drew withering criticism from liberals, for whom Justice Kavanaugh’s confirmation, amid allegations that he had assaulted a woman in the 1980s, remains one of the foremost offenses of the Trump era.

By Wednesday evening she had changed her position on the matter, saying on Twitter, “Upon further reflection and further understanding of his record, I would have voted no.”

Democrats have been frustrated by their failures to recruit some top-level targets for Senate races. They include Stacey Abrams, who ran for governor in Georgia last year but who said this year she would not run for Senate in 2020. Other potential candidates have chosen to run for president instead, including Gov. Steve Bullock of Montana; former Representative Beto O’Rourke and the former housing secretary Julián Castro, both of Texas; and former Gov. John Hickenlooper of Colorado.

But Ms. McGrath is one of several Democrats to mount a high-profile challenge to a sitting Republican senator.

Sara Gideon, the speaker of the Maine House of Representatives, said last month that she would challenge Senator Susan Collins. Jaime Harrison, the first black chairman of the South Carolina Democratic Party, has declared his candidacy for the Senate seat held by Lindsey Graham. And Mr. Kelly, the former astronaut and gun-control activist who is married to Gabrielle Giffords, said in February that he would take on Senator Martha McSally of Arizona.