For every sufficiently small $w>0$ and all sufficiently large nonsurgery times $t$, a three-dimensional Ricci flow with surgery whose surgery scale is $o(\sqrt{t})$ decomposes into a $w$-thick part and a $w$-thin part defined using the curvature scale $\rho(x,t)$. After rescaling by $t^{-1}$, every pointed sequence based at $w$-thick points has a subsequence converging smoothly on large compact subsets to a complete finite-volume hyperbolic manifold of sectional curvature $-1/4$, while the $w$-thin part is locally collapsed under a lower sectional-curvature bound.