We consider the skew-product semiflow which is generated by a scalar reaction–diffusion equation where f is uniformly almost periodic in t. The structure of the minimal set M is thoroughly investigated under the assumption that the center space $$V^c(M)$$ associated with M is no more than 2-dimensional. Such situation naturally occurs while, for instance, M is hyperbolic or uniquely ergodic. It is shown in this paper that M is a 1-cover of the hull H(f) provided that M is hyperbolic (equivalently, $$\mathrm{dim}V^c(M)=0$$ ). If $$\mathrm{dim}V^c(M)=1$$ (resp. $$\mathrm{dim}V^c(M)=2$$ with $$\mathrm{dim}V^u(M)$$ being odd), then either M is an almost 1-cover of H(f) and topologically conjugate to a minimal flow in $${\mathbb {R}}\times H(f)$$ ; or M can be (resp. residually) embedded into an almost periodically (resp. almost automorphically) forced circle-flow $$S^1\times H(f)$$ . When $$f(t,u,u_x)=f(t,u,-u_x)$$ (which includes the case $$f=f(t,u)$$ ), it is proved that any minimal set M is an almost 1-cover of H(f). In particular, any hyperbolic minimal set M is a 1-cover of H(f). Furthermore, if $$\mathrm{dim}V^c(M)=1$$ , then M is either a 1-cover of H(f) or is topologically conjugate to a minimal flow in $${\mathbb {R}}\times H(f)$$ . For the general spatially-dependent nonlinearity $$f=f(t,x,u,u_{x})$$ , we show that any stable or linearly stable minimal invariant set M is residually embedded into $${\mathbb {R}}^2\times H(f)$$ .