The density dependence of the K- optical potential V-opt(r) at zero kinetic energy is studied by fitting it to a comprehensive set of atomic data across the periodic table. Two families of solutions offering improved fits over that for a t(eff)rho(r) parameterization are identified, one containing solutions for V-opt with r.m.s. radii larger than those of the matter density rho, as expected from finite-range folding corrections. The other gives rise to Re V-opt of a size smaller than that of rho. The latter family offers substantially improved fits to the data, with particularly strong attraction inside the nucleus (200 +/- 20 MeV). For a subfamily of solutions, V-opt becomes repulsive at large distances, and solutions satisfying the low-density limit V-opt --> t((K) over bar N)rho can be found. Comments are made on the relevance of Lambda(1405) nuclear dynamics to these findings.