  The charged Higgs boson plays an essential role in distinguishing between a
wide variety of standard model extensions with multiple Higgs doublets, and has
been searched for in various collider experiments. This paper expands our
previous work to a broader Higgs mass space with discussions on subsequent
issues. We study the prospect of a light charged Higgs boson, produced by top
quark pairs at the Large Hadron Collider (LHC), and decaying into a $W$ boson
(can be off shell) and a pair of bottom quarks via on-shell production of an
intermediate neutral Higgs boson. We reinterpret the cross sections of
$WWbb\bar{b}\bar{b}$ final states measured by the ATLAS collaboration at LHC 13
TeV in the presence of the decay chain: $t \rightarrow H^+ b, H^+ \rightarrow
W^+ H_i, H_i \rightarrow b \bar{b}$, and H.c., where $H_i$ is a neutral Higgs
boson variably lighter than the charged Higgs boson. We find improved
agreements with the data and obtain limits on the total branching ratio of the
aforementioned decay chain. The limits impose the strongest constraints on the
parameter space of type-I two-Higgs-doublet model for most Higgs masses sampled
when $H_i$ is the $CP$-odd Higgs boson $A$. We also calculate potential
constraints with pseudodata in high-luminosity runs of the LHC.
