<p>Constant interpolation approximation generally yields first-order accurate methods in time, exemplified by the first-order exponential time difference (ETD1) method. Building on this approximation, this paper develops a class of three-level, simple and efficient second-order ETD methods for solving Allen-Cahn type models, including both classical and volume-conserving variants. The new ETD schemes not only maintain the concise formulation of the ETD1 method, but also preserves its advantageous properties such as the MBP preservation, discrete energy stability and volume conservation. A rigorous local third-order consistency error analysis is conducted to prove that the new ETD schemes using constant interpolation approximation achieve global second-order accuracy in time. Extensive numerical examples are presented to validate the effectiveness of the new ETD methods, including the accuracy verification, the investigation of merging of bubbles and coarsening dynamics.</p>