  We report the discovery of a new transiting exoplanet orbiting the star
TIC~257060897 and detected using {\it TESS} full frame images. We acquired
HARPS-N time-series spectroscopic data, and ground-based photometric follow-up
observations from which we confirm the planetary nature of the transiting body.
For the host star we determined: T$\rm_{eff}$=(6128$\pm$57) K,
log~g=(4.2$\pm$0.1) and [Fe/H]=(+0.20$\pm$0.04). The host is an intermediate
age ($\sim$3.5~Gyr), metal-rich, subgiant star with M$_{\star}$=(1.32$\pm$0.04)
M$_{\odot}$ and R$_{\star}$=(1.82$\pm$0.05) R$_{\odot}$. The transiting body is
a giant planet with a mass m$\rm_p=$(0.67$\pm$0.03) M$\rm_{j}$, a radius
r$\rm_p=$(1.49$\pm$0.04) R$\rm_{j}$ yielding a density $\rho_p$=(0.25$\pm$0.02)
g cm$^{-3}$ and revolving around its star every $\sim$3.66 days. TIC~257060897b
is an extreme system having one of the smallest densities known so far. We
argued that the inflation of the planet's radius may be related to the fast
increase of luminosity of its host star as it evolves outside the main sequence
and that systems like TIC~257060897b could be precursors of inflated radius
short period planets found around low luminosity red giant branch stars, as
recently debated in the literature.
