  Heavy vector-like quarks with electric charge $Q=2/3$ (also called
\textit{heavy tops}) appear naturally in many extensions of the Standard Model.
Although these typically predict the existence of further particles below the
TeV scale, direct searches for heavy tops have been performed assuming that
they decay only into SM particles. The aim of this paper is to overcome this
situation. We consider the most constraining experimental LHC searches for
vector-like quarks, including analyses of the 36 fb$^{-1}$ of data collected in
the latest run at 13 TeV of center of mass energy, as well as searches
sensitive to heavy tops decaying into a new scalar, $S$. Combining all these,
we derive bounds for arbitrary values of the heavy top branching ratios. A
simple code that automatizes this process is also provided. At the physics
level, we demonstrate that bounds on heavy tops are not inevitably weaker in
the presence of new light scalars. We find that heavy tops with masses below
$\sim 900$ GeV are excluded by direct searches, independently of whether they
decay into $Zt, Ht, Wb$ or $St$ (with $S$ giving either missing energy of
bottom quarks) or into any combination of them.
