Ascent sequences of length $n$ avoiding the pattern $021$ are enumerated by the $n$-th Catalan number $C_n=\frac{1}{n+1}\binom{2n}{n}$. In this paper, we extend this result and enumerate ascent sequences avoiding $\{021,τ\}$, where $τ$ is a pattern of length four. We in turn identify all of the corresponding Wilf-equivalence classes and find generating function formulas corresponding to each class. In a couple of cases, we make use of an auxiliary statistic and the kernel method to ascertain the generating function. In several cases, our work of enumeration is shortened by establishing the equivalence of $\{021,τ\}$- and $\{021,τ'\}$-avoiders of a given length through an explicit bijection. As a consequence of our results, one obtains new combinatorial interpretations in terms of ascent sequences for several of the entries in the OEIS.