TIC~257060897b: an inflated, low-density, hot-Jupiter transiting a rapidly evolving subgiant star.

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.