In this paper, we use local fraction derivative to show the Hölder continuity of the solution to the following nonlinear time-fractional slow and fast diffusion equation: $$\left(\partial^β+\fracν{2}(-Δ)^{α/2}\right)u(t,x) = I_t^γ\left[σ\left(u(t,x)\right)\dot{W}(t,x)\right],\quad t>0,\: x\in\mathbb{R}^d,$$ where $\dot{W}$ is the space-time white noise, $α\in(0,2]$, $β\in(0,2)$, $γ\ge 0$ and $ν>0$, under the condition that $2(β+γ)-1-dβ/α>0$. The case when $β+γ\le 1$ has been obtained in \cite{ChHuNu19}. In this paper, we have removed this extra condition, which in particular includes all cases for $β\in(0,2)$.