Let (Xt)t≥0 be a Feller process generated by a pseudo-differential operator whose symbol satisfies {norm of matrix}p({dot operator}, ξ){norm of matrix}_∞≤c(1+ξ²) and p({dot operator}, 0)≡0. We prove that, for a large class of examples, the Hausdorff dimension of the set {X_t:t∈E} for any analytic set E⊂[0, ∞) is almost surely bounded below by δ_∞dim_HE, whereδ∞{colon equals}sup{δ>0:limξ→∞infz∈RdRep(z,ξ)ξδ=∞}. This, along with the upper bound β_∞dim_HE with β∞{colon equals}inf{δ>0:limξ→∞supη≤ξsupz∈Rdp(z,η)ξδ=0} established in Böttcher, Schilling and Wang (2014), extends the dimension estimates for Lévy processes of Blumenthal and Getoor (1961) and Millar (1971) to Feller processes.