We study the continuity on the modulation spaces $$M^{p,q}$$ M p , q of Fourier multipliers with symbols of the type $$e^{i\mu (\xi )}$$ e i μ ( ξ ) , for some real-valued function $$\mu (\xi )$$ μ ( ξ ) . A number of results are known, assuming that the derivatives of order $$\ge 2$$ ≥ 2 of the phase $$\mu (\xi )$$ μ ( ξ ) are bounded or, more generally, that the second derivatives belong to the Sjöstrand class $$M^{\infty ,1}$$ M ∞ , 1 . Here we extend those results, by assuming that the second derivatives lie in the bigger Wiener amalgam space $$W(\mathcal {F}L^1,L^\infty )$$ W ( F L 1 , L ∞ ) ; in particular they could have stronger oscillations at infinity such as $$\cos |\xi |^2$$ cos | ξ | 2 . Actually our main result deals with the more general case of possibly unbounded second derivatives. In that case we have boundedness on weighted modulation spaces with a sharp loss of derivatives.