  We report the discovery and characterization of a small planet, TOI-1408 c,
on a 2.2-day orbit located interior to a previously known hot Jupiter, TOI-1408
b ($P=4.42$ d, $M=1.86\pm0.02\,M_\mathrm{Jup}$, $R=2.4\pm0.5\,R_\mathrm{Jup}$)
that exhibits grazing transits. The two planets are near 2:1 period
commensurability, resulting in significant transit timing variations (TTVs) for
both planets and transit duration variations (TDVs) for the inner planet. The
TTV amplitude for TOI-1408 c is 15% of the planet's orbital period, marking the
largest TTV amplitude relative to the orbital period measured to date.
Photodynamical modeling of ground-based radial velocity (RV) observations and
transit light curves obtained with the Transiting Exoplanet Survey Satellite
(TESS) and ground-based facilities leads to an inner planet radius of
$2.22\pm0.06\,R_\oplus$ and mass of $7.6\pm0.2\,M_\oplus$ that locates the
planet into the Sub-Neptune regime. The proximity to the 2:1 period
commensurability leads to the libration of the resonant argument of the inner
planet. The RV measurements support the existence of a third body with an
orbital period of several thousand days. This discovery places the system among
the rare systems featuring a hot Jupiter accompanied by an inner low-mass
planet.
