An empirical Bayes confidence interval has high user demand in many applications. In particular, the second-order empirical Bayes confidence interval, the coverage error of which is of the third order for large number of areas, is widely used in small area estimation when the sample size within each area is not large enough to make reliable direct estimates based on a design-based approach. Yoshimori and Lahiri (2014a) proposed a new type of confidence interval, called the second-order efficient empirical Bayes confidence interval, whose length is less than that of the direct confidence interval based on the design-based approach. However, this interval still has some disadvantages: (i) it is hard to use when at least one leverage value is high; (ii) many iterations tend to be required to obtain the estimators of one global model variance parameter as the number of areas getting larger, due to the area-specific adjustment factor. To prevent such issues, this paper proposes, as never done before, a more efficient confidence interval to allow for high leverage and reduce the number of iterations for large number of areas, by adopting a non-area-specific adjustment factor, maintaining the existing desired properties. We also reveal the relationship between the general adjustment factor and the measure of uncertainty of the empirical Bayes estimator to create a second-order confidence interval. Moreover, we present two simulation studies and a real data analysis to show the efficiency of this confidence interval.