  We report on the precise radial velocity follow-up of TOI-544 (HD 290498), a
bright K star (V=10.8), which hosts a small transiting planet recently
discovered by the Transiting Exoplanet Survey Satellite (TESS). We collected
122 high-resolution HARPS and HARPS-N spectra to spectroscopically confirm the
transiting planet and measure its mass. The nearly 3-year baseline of our
follow-up allowed us to unveil the presence of an additional, non-transiting,
longer-period companion planet. We derived a radius and mass for the inner
planet, TOI-544b, of 2.018 $\pm$ 0.076 R$_{\oplus}$ and 2.89 $\pm$ 0.48
M$_{\oplus}$ respectively, which gives a bulk density of $1.93^{+0.30}_{-0.25}$
g cm$^{-3}$. TOI-544c has a minimum mass of 21.5 $\pm$ 2.0 M$_{\oplus}$ and
orbital period of 50.1 $\pm$ 0.2 days. The low density of planet-b implies that
it has either an Earth-like rocky core with a hydrogen atmosphere, or a
composition which harbours a significant fraction of water. The composition
interpretation is degenerate depending on the specific choice of planet
interior models used. Additionally, TOI-544b has an orbital period of 1.55 days
and equilibrium temperature of 999 $\pm$ 14 K, placing it within the predicted
location of the radius valley, where few planets are expected. TOI-544b is a
top target for future atmospheric observations, for example with JWST, which
would enable better constraints of the planet composition.
