Mr. Trump needs an edge in early mail-in voting to help offset the overwhelming advantage Democrats will have once early in-person voting begins on Oct. 20. Mr. Romney’s organization on the ground in North Carolina gave him an edge over the Obama campaign in mail-in voting in 2012, helping him carry the state.

While there is too little data to point to any conclusive early-voting trends, the Clinton campaign said it was seeing encouraging signs in other states, including in the Midwest, where Mr. Trump is hoping for a surge of blue-collar support. In Ohio, where polls once showed Mr. Trump leading, the race has tightened recently. Absentee-ballot requests from heavily Democratic Cuyahoga County, home to Cleveland, accounted for one out of every six requests statewide, Mr. Mook said. In 2012, 41 percent of voters in the county voted absentee, and that figure is expected to climb this year.

But the Republican National Committee’s investment in building a more sophisticated voter identification and registration program has been helping the party make up for lost time. Teams have fanned out across swing states, with clipboard-carrying staff members hitting Department of Motor Vehicles lines in Las Vegas, Hispanic evangelical churches in Florida and train stations in Virginia.

This is a change from years in which the party failed to create an on-the-ground network that could compete with the one the Obama campaign began working on in 2007. Adding to the challenge, the Republican Party has had to assume the work for the Trump campaign, which has neglected to undertake the kind of voter outreach that Mr. Romney and other nominees did.

“We’ve both learned that you have to change the electorate,” said Mr. Young, the Republican National Committee field director. “That’s what Obama did so well, and that’s what we are doing now.”

But Republicans are not seeing any clear, heartening trends in the early voting. In Iowa, for example, ballot requests from registered Democrats are down significantly more than requests from Republicans — but after the first debate, Mr. McDonald of the University of Florida found that ballot requests among Republicans there fell further, something he said could be attributable to Mr. Trump’s widely panned performance.

Registered Republicans in Florida are also ahead of Democrats in the number of mail-in ballots requested — but by a margin of just 81,000. In Nevada, there are almost 23,000 more registered Republicans than there were in 2012, while Democrats have added only about 4,000.