Missing or incomplete outcome data is a ubiquitous problem in biomedical and social sciences. Under the missing at random setup, inverse probability weighting is widely applied to estimate and make inference on the population objects of interest, but it is known that its performance can be poor in practical sample sizes. Recently, to overcome this problem, several alternative weighting methods have been proposed that directly balance the distributional characteristics of covariates. These existing balancing methods are useful for obtaining point estimates of the population objects. The purpose of this paper is to develop a new weighting scheme, based on Empirical Likelihood, that would be useful for conducting interval estimation or hypothesis testing. In particular, we propose re-weighting the covariate balancing weights so that the resulting objective function admits an asymptotic chi-square calibration. Our re-weighting method is naturally extended to inference on treatment effects, data combination models, and high-dimensional covariates. Simulation and empirical examples illustrate usefulness of the proposed method.