Since the waiting-list scandal broke last year, the department has broadly expanded access to care. Its doctors and nurses have handled 2.7 million more appointments than in any previous year, while authorizing 900,000 additional patients to see outside physicians. In all, agency officials say, they have increased capacity by more than seven million patient visits per year — double what they originally thought they needed to fix shortcomings.

But what was not foreseen, department leaders say, was just how much physician workloads and demand from veterans would continue to soar — by one-fifth, in fact, at some major veterans hospitals over just the past year.

According to internal department budget documents obtained by The New York Times, physician workloads — as measured by an internal metric known as “relative value units” — grew by 21 percent at hospitals and clinics in the region that includes Alabama, Georgia and South Carolina; by 20 percent in the Southern California and southern Nevada regions; and by 18 percent in North Carolina and Virginia. And by the same measure, physician care purchased for patients treated outside the department grew by 50 percent in the region encompassing Pennsylvania and by 36 percent in the region that includes Michigan and Indiana.

Those data include multiple appointments by individual patients and reflect the fact that patients typically now schedule more appointments than they did in the past. But even measured by the number of individuals being treated, the figures are soaring in many places: From 2012 to 2014, for example, the number of patients receiving treatment grew by 18 percent at the Las Vegas medical center; by 16 percent in Hampton, Va.; and by 13 percent in Fayetteville, N.C., and Portland, Ore.

Mr. Gibson said in the interview that officials had been stunned by the number of new patients seeking treatment even as the V.A. had increased its capacity. He said he was frustrated that the agency was running short of funds. “We have been pushing to accelerate access to care for veterans, but where we now find ourselves is that if we don’t do something different we’re going to be $2.7 billion short,” he said.