We consider phaseless inverse scattering for the multidimensional Schr\"odinger equation with unknown potential $v$ using the method of known background scatterers. In particular, in dimension $d \geq 2$, we show that $|f_1|^2$ at high energies uniquely determines $v$ via explicit formulas, where $f_1$ is the scattering amplitude for $v+w_1$, $w_1$ is an a priori known nonzero background scatterer, under the condition that $supp\, v$ and $supp\, w_1$ are sufficiently disjoint. If this condition is relaxed, then we give similar formulas for finding $v$ from $|f|^2 , |f_1|^2$, where $f$ is the scattering amplitude for $v$. In particular, we continue studies of [Novikov, J. Geom. Anal. 26(1), 346–359, 2016], [Leshem et al, Nature Communications 7(1), 1–6, 2016].