In this paper, we derive improved a priori error estimates for families of hybridizable interior penalty discontinuous Galerkin (H-IP) methods using a variable penalty for second-order elliptic problems. The strategy is to use a penalization function of the form O(1/h^{1+δ}), where h denotes the mesh size and δ is a user-dependent parameter. We then quantify its direct impact on the convergence analysis, namely, the (strong) consistency, discrete coercivity and boundedness (with h δ-dependency), and we derive updated error estimates for both discrete energy-and L^2-norms. All theoretical results are supported by numerical evidence.