For standard $p$-functions, an upper bound for $M = p(1)$, for a given value $m$ of $m(p) = \min\{p(t), 0 < t \leqq 1\}$, was proved in a previous paper by the author. The bound implied that $\nu_0 \leqq .590, \nu_0$ being the constant defined by $$I_M = \inf\{m(p)|p(1) = M\},\quad \nu_0 = \inf\{M|I_M > 0\},$$ in which $p$ varies over the class of standard $p$-functions. In the present paper both of these upper bounds are sharpened by a refinement of the argument, the limit for $\nu_0$ being reduced to .560.