In this paper, we consider a Wigner matrix $A$ with entries whose cumulative distribution decays as $x^{-\alpha}$ with $2<\alpha<4$ for large $x$. We prove that the fluctuations of the linear statistics $N^{-1}\operatorname{Tr} \varphi(A)$, for some nice test functions $\varphi$, have order $N^{-\alpha/4}$. The behavior of such fluctuations has been understood for both heavy-tailed matrices (i.e. $\alpha < 2$) and light-tailed matrices (i.e. $\alpha > 4$). This paper fills in the gap of understanding for $2 < \alpha < 4$. We find that while linear spectral statistics for heavy-tailed matrices have fluctuations of order $N^{-1/2}$ and those for light-tailed matrices have fluctuations of order $N^{-1}$, the linear spectral statistics for half-heavy-tailed matrices exhibit an intermediate $\alpha$-dependent order of $N^{-\alpha/4}$.