  The class of $2K_2$-free graphs has been well studied in various contexts in
the past. In this paper, we study the chromatic number of $\{butterfly,
hammer\}$-free graphs, a superclass of $2K_2$-free graphs and show that a
connected $\{butterfly, hammer\}$-free graph $G$ with $\omega(G)\neq 2$ admits
$\binom{\omega+1}{2}$ as a $\chi$-binding function which is also the best
available $\chi$-binding function for its subclass of $2K_2$-free graphs. In
addition, we show that if $H\in\{C_4+K_p, P_4+K_p\}$, then any $\{butterfly,
hammer, H\}$-free graph $G$ with no components of clique size two admits a
linear $\chi$-binding function. Furthermore, we also establish that any
connected $\{butterfly, hammer, H\}$-free graph $G$ where $H\in \{(K_1\cup
K_2)+K_p, 2K_1+K_p\}$, is perfect for $\omega(G)\geq 2p$.
