Given initial data $$u_0=(u_0^{\mathrm{h}},u_0^3)\in H^{\frac{1}{2}}({{\mathbb {R}}}^3)\cap B^{0,\frac{1}{2}}_{2,1}({{\mathbb {R}}}^3)$$ with $$u^{{\mathrm{h}}}_0$$ belonging to $$L^2({{\mathbb {R}}}^3)\cap L^\infty ({{\mathbb {R}}}_{\mathrm{v}}; H^{-\delta }({{\mathbb {R}}}^2_{\mathrm{h}}))\cap L^\infty ({{\mathbb {R}}}_{\mathrm{v}}; H^3({{\mathbb {R}}}^2_{\mathrm{h}}))$$ for some $$\delta \in ]0,1[,$$ if in addition $$\partial _3u_0$$ belongs to the homogeneous anisotropic Sobolev space, $$H^{-\frac{1}{2},0},$$ we prove that the classical 3-D Navier–Stokes system has a unique global Fujita–Kato solution provided that the $$H^{-\frac{1}{2},0}$$ norm of $$\partial _3u_0$$ is sufficiently small compared to $$\exp \left( -\,C\bigl (A_\delta (u^{{\mathrm{h}}}_0)+B_\delta (u_0)\bigr )\right) $$ with $$A_\delta (u^{{\mathrm{h}}}_0)$$ and $$B_\delta (u_0)$$ being scaling invariant quantities of the initial data, which is scaling invariant with respect to the variable $$x_3$$. This result provides some classes of large initial data which are large in Besov space $$B^{-1}_{\infty ,\infty }$$ and which generate unique global solutions to the 3-D Navier–Stokes system. In particular, we extend the previous results in [5,7,10] for initial data with a slow variable to multi-scales slow variable initial data.