  Let $p$ be a prime number. Continuing and extending our previous paper with
the same title, we prove explicit rates of overconvergence for modular
functions of the form $\frac{E_k^{\ast}}{V(E_k^{\ast})}$ where $E_k^{\ast}$ is
a classical, normalized Eisenstein series on $\Gamma_0(p)$ and $V$ the $p$-adic
Frobenius operator. In particular, we extend our previous paper to the primes
$2$ and $3$. For these primes our main theorem improves somewhat upon earlier
results by Emerton, Buzzard and Kilford, and Roe. We include a detailed
discussion of those earlier results as seen from our perspective. We also give
some improvements to our earlier paper for primes $p\ge 5$.
  Apart from establishing these improvements, our main purpose here is also to
show that all of these results can be obtained by a uniform method, i.e., a
method where the main points in the argumentation is the same for all primes.
  We illustrate the results by some numerical examples.
