  In models where the Higgs is realized as a pseudo-Nambu-Goldstone boson
(pNGB) of some global symmetry breaking, there are often remaining pNGBs of
some $U(1)$ groups (called `pseudo-axions'), which could lead to smoking gun
signatures of such scenarios and provide important clues on the electroweak
symmetry breaking mechanism. As a concrete example, we investigate the
phenomenology of the pseudo-axion in the anomaly-free Simplest Little Higgs
(SLH) model. After clarifying a subtle issue related to the effect of symmetric
vector-scalar-scalar (VSS) vertices (e.g.
$Z_\mu(H\partial^\mu\eta+\eta\partial^\mu H)$), we show that for natural region
in the parameter space, the SLH pseudo-axion is top-philic, decaying almost
exclusively to a pair of top quarks. The direct and indirect (i.e. via heavy
particle decay) production of such a pseudo-axion at the $14\,\mbox{TeV}$
(HL-)LHC turn out to suffer from either large backgrounds or small rates,
making its detection quite challenging. A $pp$ collider with higher energy and
luminosity, such as the $27\,\mbox{TeV}$ HE-LHC, or even the $100\,\mbox{TeV}$
FCC-hh or SppC, is therefore motivated to capture the trace of such a pNGB.
