  The study of neutrino non-standard interactions (NSI) is a well-motivated
phenomenological scenario to explore new physics beyond the Standard Model. The
possible scalar coupling of neutrinos ($\nu$) with matter is one of such new
physics scenarios that appears as a sub-dominant effect that can impact the
$\nu$-oscillations in matter. The presence of scalar NSI introduces an
additional contribution directly to the $\nu$-mass matrix in the interaction
Hamiltonian and subsequently to the $\nu$-oscillations. This indicates that
scalar NSI may have a significant impact on measurements related to
$\nu$-oscillations e.g. leptonic CP phase $(\delta_{CP})$, $\theta_{23}$ octant
and neutrino mass ordering (MO). The linear scaling of the effects of scalar
NSI with matter density also motivates its exploration in long-baseline (LBL)
experiments. In this paper, we study the impact of a scalar-mediated NSI on the
MO sensitivity of DUNE, HK and HK+KNO, which are upcoming LBL experiments. We
study the impact on MO sensitivities at these experiments assuming that scalar
NSI parameters are present in nature and is known from other non-LBL
experiments. We observe that the presence of diagonal scalar NSI elements can
significantly affect the $\nu$-mass ordering sensitivities. We then also
combine the data from DUNE with HK and HK+KNO to explore possible synergy among
these experiments in a wider parameter space. We also observe a significant
enhancement in the MO sensitivities for the combined analysis.
