Among iron chalcogenide superconductors, FeS can be viewed as a simple,
highly compressed relative of FeSe without nematic phase and with weaker
electronic correlations. Under pressure, however, the superconductivity of
stoichiometric FeS disappears and reappears, forming two domes. We perform
electronic structure and spin fluctuation theory calculations for tetragonal
FeS in order to analyze the nature of the superconducting order parameter. In
the random phase approximation we find a gap function with d-wave symmetry at
ambient pressure, in agreement with several reports of a nodal superconducting
order parameter in FeS. Our calculations show that, as a function of pressure,
the superconducting pairing strength decreases until a Lifshitz transition
takes place at 4.6 GPa. As a hole pocket with a large density of states appears
at the Lifshitz transition, the gap symmetry is altered to sign-changing
s-wave. At the same time the pairing strength is severely enhanced and
increases up to a new maximum at 5.5 GPa. Therefore, our calculations naturally
explain the occurrence of two superconducting domes in FeS.