John Eastman, the conservative lawyer whose plan to block congressional certification of the 2020 election failed in spectacular fashion on Jan. 6, 2021, sent an email two weeks later arguing that pro-Trump forces should sue to keep searching for the supposed election fraud he acknowledged they had failed to find.

On Jan. 20, 2021, hours after President Biden’s inauguration, Mr. Eastman emailed Rudolph W. Giuliani, former President Donald J. Trump’s personal lawyer, proposing that they challenge the outcome of the runoff elections in Georgia for two Senate seats that had been won on Jan. 5 by Democrats.

“A lot of us have now staked our reputations on the claims of election fraud, and this would be a way to gather proof,” Mr. Eastman wrote in the previously undisclosed email, which also went to others, including a top Trump campaign adviser. “If we get proof of fraud on Jan. 5, it will likely also demonstrate the fraud on Nov. 3, thereby vindicating President Trump’s claims and serving as a strong bulwark against Senate impeachment trial.”

The email, which was reviewed by The New York Times and authenticated by people who worked on the Trump campaign at the time, is the latest evidence that even some of Mr. Trump’s most fervent supporters knew they had not proven their baseless claims of widespread voting fraud — but wanted to continue their efforts to delegitimize the outcome even after Mr. Biden had taken office.