  We establish several mixed $A_p$-$A_\infty$ bounds for Calder\'on-Zygmund
operators that only involve one supremum. We address both cases when the
$A_\infty$ part of the constant is measured using the exponential-logarithmic
definition and using the Fujii-Wilson definition. In particular, we answer a
question of the first author and provide an answer, up to a logarithmic factor,
of a conjecture of Hyt\"onen and Lacey. Moreover, we give an example to show
that our bounds with the logarithmic factors can be arbitrarily smaller than
the previously known bounds (both one supremum and two suprema).
