<p>In this paper, we present some enhanced error estimates for augmented subspace method with the nonconforming Crouzeix-Raviart (CR) element. Before the novel estimates, we derive the explicit error estimates for the cases of single eigenpair and multiple eigenpairs based on our defined spectral projection operators, respectively. Subsequently, we first rigorously prove that the augmented subspace approach based on CR elements demonstrates the second-order convergence rate with respect to the coarse space between the two steps of augmented subspace iteration. This rate aligns with the results of actual numerical experiments. The augmented subspace method’s second-order algebraic error estimates clearly show how the coarse space affects the algebraic error’s convergence rate, offering fresh perspectives on the method’s effectiveness. Numerical experiments are finally supplied to verify these new estimate results and the efficiency of our algorithms.</p>