Suppose $j_N(τ)$ and $j_N^{*}(τ)$ are the Hauptmoduln of the congruence subgroup $Γ_0(N)$ and the Fricke group $Γ^{*}_0(N)$, respectively. In [7], the authors predicted that, like Klein's $j$-function, the Fourier coefficients of $j_N(τ)$ and $j_{N}^{*}(τ)$ in some arithmetic progression are both even and odd with density $\frac{1}{2}$. In this article, we can find some arithmetic progression of $n$ where the Fourier coefficients of $j_6(τ)$ (resp. $j_6^{*}(τ)$ and $j_{10}(τ)$) are almost always even. Furthermore, using Hecke eigenforms and Rogers-Ramanujan continued fraction, we obtain infinite families of congruences for $j_6(τ)$, $j_6^{*}(τ)$, $j_{10}(τ),$ and $j_{10}^{*}(τ)$.