We report on an extensive experimental study of the low-temperature--high-field corner of the $H\text{\ensuremath{-}}T$ phase diagram in FeSe using high-sensitivity specific heat $C$ measurements. Indeed, the superconducting gap and Fermi and Zeeman energies are surprisingly close in this compound, possibly leading to the existence of different superconducting phases as well as Lifshitz transitions in the electronic structure. The nature of this part of the phase diagram hence remains debated, and we show here that two distinct anomalies are visible in $C(H)$ (below $\ensuremath{\sim}3$ K): a (smeared) jump at the superconducting transition and a kink in $C(H)$. This second structure lies either below (for $H$ close to the $ab$ plane) or above (for field orientations close to the $c$ direction) the superconducting transition, indicating that it is most probably related to a field-induced change in the electronic structure. Moreover, quantum oscillations are clearly observed below $\ensuremath{\sim}2$ K, and $C(H)$ can be well described by the Lifshitz-Kosevich formula for fields both above and below the kink (with an effective mass ${m}^{*}\ensuremath{\sim}4{m}_{e}$ and a frequency $F\ensuremath{\sim}200$ T, in good agreement with the values previously inferred from magnetotransport measurements for the hole sheet).