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We report the discovery of a new transiting exoplanet orbiting the starTIC~257060897 and detected using {\it TESS} full frame images. We acquiredHARPS-N time-series spectroscopic data, and ground-based photometric follow-upobservations 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$0.04). The host is an intermediateage ($\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}$. The transiting body isa giant planet with a mass m$\rm_p=$(0.67$\pm$0.03) M$\rm_{j}$, a radiusr$\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~257060897bis an extreme system having one of the smallest densities known so far. Weargued that the inflation of the planet's radius may be related to the fastincrease of luminosity of its host star as it evolves outside the main sequenceand that systems like TIC~257060897b could be precursors of inflated radiusshort period planets found around low luminosity red giant branch stars, asrecently debated in the literature.

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