  In this paper we use a modification of the Newtonian gravitational potential
with a non-linear Yukawa-like correction, as it was proposed by C. Will earlier
to obtain new bounds on graviton mass from the observed orbits of S-stars
around the Galactic Center (GC). This phenomenological potential differs from
the gravitational potential obtained in the weak field limit of Yukawa gravity,
which we used in our previous studies. We also assumed that the orbital
precession of S-stars is close to the prediction of General Relativity (GR) for
Schwarzschild precession, but with a possible small discrepancy from it. This
assumption is motivated by the fact that the GRAVITY Collaboration in 2020 and
in 2022 detected Schwarzschild precession in the S2 star orbit around the
Supermassive Black Hole (SMBH) at the GC. Using this approach, we were able to
constrain parameter $\lambda$ of the potential and, assuming that it represents
the graviton Compton wavelength, we also found the corresponding upper bound of
graviton mass. The obtained results were then compared with our previous
estimates, as well as with the estimates of other authors.
