We examine the limiting free energy density $$a(\varrho ,0 + ) \equiv \mathop {\lim }\limits_{\gamma \to 0} a(\varrho ,\gamma )$$ of a classical system of particles with the two-body potentialq(r)+γ v K (γr), at density ϱ inv dimensions. Starting from a variational formula fora(ϱ, 0 + ), obtained in Part I of these papers, we obtain a new upper bound ona(ϱ, 0 + ) given by $$a(\varrho ,0 + ) \mathbin{\lower.3ex\hbox{$\buildrel<\over{\smash{\scriptstyle=}\vphantom{_x}}$}} CE\{ ME[a^0 (\varrho ) + {\textstyle{1 \over 4}}\tilde K_{\min } \varrho ^2 ] + ({\textstyle{1 \over 2}}\alpha - {\textstyle{1 \over 4}}\tilde K_{\min } )\varrho ^2 \} $$ . HereM E f, called the mid-point envelope off, is defined for any functionf by $$MEf(\varrho ) \equiv \mathop {\inf }\limits_h {\textstyle{1 \over 2}}[f(\varrho + h) + f(\varrho - h)];$$ C E f, called the convex envelope off, is defined for anyf as the maximal convex function not exceedingf; also α ≡ ed s K(s) and $$\tilde K_{\min } $$ is the minimum of the Fourier transform ofK, whilea 0(ϱ) is the free energy density forK = 0. For the class of functionsK such that $$\tilde K_{\min } $$ < 0 and $$\tilde K_{\min } $$ <2α, we deduce from this upper bound thata(ϱ, 0 + )