We prove that the hydrodynamic pressure p associated to the velocity u ∈ C^θ (Ω), θ ∈ (0, 1), of an inviscid incompressible fluid in a bounded and simply connected domain Ω ⊂ R d with C 2+ boundary satisfies p ∈ C^θ (Ω) for θ ≤ 1 2 and p ∈ C^{1,2θ−1} (Ω) for θ > 1 2. This extends the recent result of [4] obtained in the planar case to every dimension d ≥ 2 and it also doubles the pressure regularity for θ > 1 2. Differently from [4], we do not introduce a new boundary condition for the pressure, but instead work with the natural one. In the boundary-free case of the d-dimensional torus, we show that the double regularity of the pressure can be actually achieved under the weaker assumption that the divergence of the velocity is sufficiently regular, thus not necessarily zero.