  We consider the defocusing nonlinear wave equation $\Box u = \lvert u
\rvert^{p-1} u$ in $\mathbb{R}^3 \times[0,\infty)$. We prove that for any
initial datum with a scaling-subcritical norm bounded by $M_0$ the equation is
globally well-posed for $p=5+\delta$ where $\delta \in (0,\delta_0(M_0))$.
