Fix $\varepsilon>0$, $A>0$, an integer $k\geq 10$, and $T<\infty$. There exists $r=r(\varepsilon,A,k,T,g(0))>0$ with the following property. Let $(M^3,g(t))$, $t \in [0,T)$, be a Ricci flow on a closed three-manifold satisfying the conclusion of the Canonical Neighborhood Theorem with parameters $(\varepsilon,A,k)$ at all points with scalar curvature at least $r^{-2}$. Interpret that conclusion in the following form: each such point is the center of an $\varepsilon$-neck, lies in an $\varepsilon$-cap whose non-core part consists of centers of $\varepsilon$-necks, or lies in a compact positively curved component. Suppose that $R(x,t) \geq r^{-2}$. If $x$ is not contained in the core of an $\varepsilon$-cap and is not in a compact positively curved component, then a neighbourhood of $x$ at time $t$ is an $\varepsilon$-neck.