The first paper in this series introduced a new approach to strong convergence of random matrices that is based primarily on soft arguments. This method was applied to achieve a refined qualitative and quantitative understanding of strong convergence of random permutation matrices and of more general representations of the symmetric group. In this paper, we introduce new ideas that make it possible to achieve stronger quantitative results and that facilitate the application of the method to new models. When applied to the Gaussian GUE/GOE/GSE ensembles of dimension $N$, these methods achieve strong convergence for noncommutative polynomials with matrix coefficients of dimension $\exp(o(N))$. This provides a sharp form of a result of Pisier on strong convergence with coefficients in a subexponential operator space. Analogous results up to logarithmic factors are obtained for Haar-distributed random matrices in $\mathrm{U}(N)/\mathrm{O}(N)/\mathrm{Sp}(N)$. We further illustrate the methods of this paper in the following applications. 1. We obtain improved rates for strong convergence of random permutations. 2. We obtain a quantitative form of strong convergence of the model introduced by Hayes for the solution of the Peterson-Thom conjecture. 3. We prove strong convergence of tensor GUE models of $Γ$-independence. 4. We prove strong convergence of irreducible representations of $\mathrm{U}(N)$ of dimension up to $\exp(N^{1/3-δ})$, improving a result of Magee and de la Salle.