In the 2018 fiscal year, 1.7 million veterans used some form of private care. That number, based on the department’s eligibility projections, could increase as much as 30 percent under the new Mission Act, adding just over a half-million veterans to the pool seeking private care — although both critics and supporters of the change believe that number is low.

“I think it’s safe to assume that over the next several years you will see more veterans getting their health care in the community,” said Dan Caldwell, a senior adviser to Concerned Veterans for America — an advocacy group with ties to the billionaire industrialist Charles G. Koch — which has pursued expanding the use of private health care for veterans. Democrats on Capitol Hill agree.

Even so, it will take some time to see whether the expanded access to private care results in a gentle migration of veterans who are not being well served by nearby V.A. centers, or in a shift of a significant share of veteran care to an increasingly expensive private health care marketplace, starving the V.A., as critics have warned. The quality of care in the private system, as well as wait times within it, will be closely monitored by both opponents and supporters of private care.

“I don’t expect a flood of veterans coming in tomorrow or next week saying I want to go into the private sector,” Robert Wilkie, the secretary of veterans affairs, said Tuesday in an interview. “I’m confident that we will have a good rollout. I’m looking forward to it.” Mr. Wilkie said he will visit the V.A. in Salem, Va., on Thursday to observe the start there.

Mr. Wilkie has repeatedly said that most veterans would continue to seek care at department health care centers because they largely like the care and the understanding of veterans’ issues. Several studies underscore the V.A.’s popularity.