  We prove sufficient conditions for the boundedness and compactness of
Toeplitz operators $T_a$ in weighted sup-normed Banach spaces $H_v^\infty$ of
holomorphic functions defined on the open unit disc $\mathbb{D}$ of the complex
plane; both the weights $v$ and symbols $a$ are assumed to be radial functions
on $\mathbb{D}$. In an earlier work by the authors it was shown that there
exists a bounded, harmonic (thus non-radial) symbol $a$ such that $T_a$ is not
bounded in any space $H_v^\infty$ with an admissible weight $v$. Here, we show
that a mild additional assumption on the logarithmic decay rate of a radial
symbol $a$ at the boundary of $\mathbb{D} $ guarantees the boundedness of
$T_a$.
  The sufficient conditions for the boundedness and compactness of $T_a$, in a
number of variations, are derived from the general, abstract necessary and
sufficient condition recently found by the authors. The results apply for a
large class of weights satisfying the so called condition$(B)$, which includes
in addition to standard weight classes also many rapidly decreasing weights.
