Using scattering matrix formalism, we report the formation of a special hidden singular line connecting multiple second-order hidden singular points, in a non-uniformly pumped degenerate optical microcavity. Such singularities are known as exceptional points (EPs), and the line is proposed as an exceptional line. Exploring the unconventional behavior of cavity resonances created by a spatially imbalanced gain-loss profile, we have established the adiabatic state-flipping mechanism of coupled resonances encountering such EPs. Various interesting encircling situations, incorporating smooth as well as fluctuating variations of the control parameters, have been analyzed. We exploit the above scheme for the first time, to the best of our knowledge, to analyze the optical performance and stability of the cascaded flip-of-state phenomenon assisted by successive encirclement of either single or multiple singularities following the exceptional line in the context of optical mode converters. © 2017 Optical Society of America.