In this study, we present a new class of quasi-interpolatory non-stationary Hermite subdivision schemes reproducing exponential polynomials. This class extends and unifies the well-known Hermite schemes, including the interpolatory schemes. Each scheme in this family has tension parameters which provide design flexibility, while obtaining at least the same or better smoothness compared to an interpolatory scheme of the same order. We investigate the convergence and smoothness of the new schemes by exploiting the factorization tools of non-stationary subdivision operators. Moreover, a rigorous analysis for the approximation order of the non-stationary Hermite scheme is presented. Finally, some numerical results are presented to demonstrate the performance of the proposed schemes. We find that the quasi-interpolatory scheme can circumvent the undesirable artifacts appearing in interpolatory schemes with irregularly distributed control points. (C) 2019 Elsevier Inc. All rights reserved.