Six Co-substituted magnetites CoxFe3-xO4, with x less-than-or-equal-to 0.04 have been studied by Fe-57 Mossbauer spectroscopy at temperatures T between their respective Verwey transition and almost-equal-to 900 K. The Curie temperature T(C) decreases from 868 K for x =0 to 859 K for x =0.03. All spectra were decomposed into tetrahedral (A-site) and octahedral (B-site) components. The obtained results are consistent with the fact that at low temperatures above the Verwey transition, the spins lie along a domain's [100] axis, but flip to a [111] axis at a temperature strongly increasing with increasing x. The temperature variations of the hyperfine parameters (center shift delta, quadrupole shift epsilon(Q), magnetic hyperfine field H(hf) and dipolar field H(dip)) have been determined. The delta and H(hf) values for the octahedral Fe are consistent with the well-known fast electron exchange between the involved Fe2+ and Fe3+ species. From the temperature dependence of the B-site linewidth it is concluded that the nature of the exchange process at relatively low temperatures, depending on x, is different from that at higher temperatures. The epsilon(Q) vs T and H(dip) vs T curves for the B sites of the x = 0, 0.005, and 0.01 samples (both mentioned parameters for the A sites are zero) were interpreted on the basis of a common crystalline-field model, yielding, as argued, unrealistic values for certain involved physical quantities. The behavior of the center shifts implies a slight temperature variation for the intrinsic isomer shift. Unlike the A-site hyperfine fields, the temperature dependence of the B-site fields could not be reproduced adequately by the usual formulas based on Heisenberg-type exchange. Instead, a non-localized-electron model led to an excellent description of the experimental curves. The bandwidth, which appears in that model, turned out to be high (0.82 eV for x =0 at 300 K) and is suggested to be responsible for the failure of the crystalline-field model to explain the temperature variation of e2qQ/2 and H(dip). For all compositions epsilon(Q) and H(dip) at selected temperatures have also been determined from spectra recorded in external magnetic fields. These spectra further indicate that a field-induced contribution to the total hyperfine field on the B sites (not on A sites) is to be taken into account. Its origin is discussed briefly.