Let \(G\) be a finite group with Frattini subgroup \(\mathrm{Frat}(G)\). The intersection number \(\alpha (G)\) is defined to be the least number of maximal subgroups of \(G\) whose intersection is \(\mathrm{Frat}(G)\) and the minimal base size \(\beta (G)\) is the size of the smallest base for a faithful primitive action of \(G\). More precisely, if \(H\) is maximal in \(G\), then \(b(G,H)\) denotes the size of the smallest base for the primitive permutation action of \(G\) on the cosets of \(H\). In earlier work [J. Comb. Theory, Ser. A 171, Article ID 105175, 32 p. (2020; Zbl 1477.20028)], the authors determined the best possible bounds on \( \alpha (G)\) and \(\beta (G)\) for simple groups and in the present paper they extend their results to almost simple groups. (Theorem 1) If \(G\) is almost simple then \(\alpha (G)\leq 4\) with equality if and only if \(G\cong U_{4}(2).2\) and \(\beta (G)\leq 4\) with equality if and only if \(G\cong S_{6}\) or the socle \(\mathrm{soc}(G)\cong U_{4}(2)\). (Theorem 2) If \(a\geq b\geq 2\) with \( ab>4\) then \(b(S_{ab},S_{b}\wr S_{a})=3\) except when \((a,b)=(3,2)\) or \(b\geq 3 \) and \(a\geq \max (b+3,8)\) (when it is \(4\) and \(2\), respectively). (Theorem 3) If \(G\) is a finite soluble group, then \(\alpha (G)\) is bounded by the length of a chief series for \(G\) and if \(G^{\prime }\) is nilpotent then \( \alpha (G)\) is bounded by the number of chief factors not contained in \(\mathrm{Frat}(G)\). Theorem 4 gives a general bound on \(\alpha (G)\) for an arbitrary finite group.