By offering an unexpected grace period to businesses with between 50 and 99 employees, administration officials are hoping to defuse another potential controversy involving the 2010 health-care law, which has become central to Republicans’ campaign to make political gains in this year’s midterm election.

Even the nation’s largest employers got a significant concession: They can avoid a fine by offering coverage to 70 percent of their full-time employees in 2015 and 95 percent starting in 2016.