  The $U$-spin symmetry provides a powerful tool to extract the angle $\gamma$
of the Unitarity Triangle and the $B_s^0$-$\bar{B}_s^0$ mixing phase $\phi_s$
from CP violation in the $B^0_s\to K^-K^+$, $B^0_d\to\pi^-\pi^+$ system. LHCb
has obtained first results with uncertainties at the $7^\circ$ level. Due to
$U$-spin-breaking corrections, it will be challenging to reduce the uncertainty
below ${\cal O}(5^\circ)$ at Belle II and the LHCb upgrade. We propose a new
strategy, using $\gamma$ as input and utilizing $B^0_s\to K^-\ell^+\nu_\ell$,
$B^0_d\to \pi^-\ell^+\nu_\ell$ decays, which allows an extraction of $\phi_s$
with a future theoretical precision of up to ${\cal O}(0.5^\circ)$, thereby
matching the experimental prospects. Since $B^0_s\to K^-K^+$ is dominated by
penguin topologies, new sources of CP violation may be revealed.
