In the high-energy $p+p$ collisions, the fundamental processes of extensive air shower development and hence the origin of the ultra-high energy cosmic ray can be studied by measuring the cross section of the very forward particles. On the other hand, transverse single spin asymmetry ($A_{\scriptsize{\textrm{N}}}$) for very forward particle production is an unique tool to understand the spin-involved diffractive particle production mechanism. $A_{\scriptsize{\textrm{N}}}$ for very forward neutron production has been interpreted by an interference between $\pi$ (spin flip) and $a_1$ (spin non-flip) exchange. Since the neutron $A_{\scriptsize{\textrm{N}}}$ has been studied only in a narrow transverse momentum ($p_{\scriptsize{\textrm{T}}}$) range of $p_{\scriptsize{\textrm{T}}} < 0.4$ GeV/$c$, the RHICf experiment measured the neutron $A_{\scriptsize{\textrm{N}}}$ in a wide $p_{\scriptsize{\textrm{T}}}$ coverage up to $\sim$1 GeV/$c$ to test the validity of the $\pi$ and $a_1$ exchange model in the higher $p_{\scriptsize{\textrm{T}}}$ region. In this poster, we report our analysis status and result on the neutron $A_{\scriptsize{\textrm{N}}}$ measured by the RHICf experiment.