The authors address the question of local convergence rate of conservative Lip+-stable approximations uepsilon(x,t) to the entropy solution u(x,t) of a genuinely nonlinear conservation law. This paper extends the previous results by including lip+ -unbounded initial data. Specifically, it is shown that for arbitrary LINF intersection BV initial data, u and its derivatives may be recovered with an almost optimal, modulo a spurious log factor, error of O(ε/Inε/). This analysis relies on obtaining new Lip+-stability estimates for the speed a(uε), rather than for uepsilon itself. This enables the establishment of an O(ε/Inε) convergence rate in W-1,1, which, in turn, implies the above mentioned local convergence rate.