In coordinate-transformation-based approaches to electromagnetic concealment (``cloaking’’) of objects, use of higher-order (quadratic) mappings has been proposed as an effective device to obtain satisfactory responses without the use of magnetic materials that are complicated to synthesize at optical frequencies. In this article, we explore a new higher-order algebraic transformation, which allows, in principle, for a broader range of applicability and further parametric optimization. Via full-wave numerical studies of near- and far-field observables, we assess its performance by comparison with various reference cases (nonreduced parameters, quadratic transformation, no cloak).