  The minimal $U(1)_{L_\mu-L_\tau}$ gauge symmetry extended Standard Model (SM)
is a well motivated framework that resolves the discrepancy between the
theoretical prediction and experimental observation of muon anomalous magnetic
moment. We envisage the possibility of identifying the beyond Standard Model
Higgs of $U(1)_{L_\mu-L_\tau}$ sector, non-minimally coupled to gravity, as the
inflaton in the early universe, while being consistent with the $(g-2)_\mu$
data. Although the structure seems to be trivial, we observe that taking into
consideration of a complete cosmological history starting from inflation
through the reheating phase to late-time epoch along with existing constraints
on $U(1)_{L_\mu-L_\tau}$ model parameters leave us a small window of allowed
reheating temperature. This further results into restriction of $(n_s-r)$ plane
which is far severe than the one in a generic non-minimal quartic inflationary
set up.
