  Besides the yield ratio of free spectator neutrons produced in ultracentral
$^{96}$Zr+$^{96}$Zr to $^{96}$Ru+$^{96}$Ru collisions, we propose that the
yield ratio $N_n/N_p$ of free spectator neutrons to protons in a single
collision system at RHIC and LHC can be a more sensitive probe of the
neutron-skin thickness $\Delta r_{\mathrm{np}}$ and the slope parameter $L$ of
the symmetry energy. The idea is demonstrated based on the proton and neutron
density distributions of colliding nuclei obtained from
Skyrme-Hartree-Fock-Bogolyubov calculations, and a Glauber model that provides
information of spectator matter. The final spectator particles are produced
from direct emission, clusterization by a minimum spanning tree algorithm or a
Wigner function approach, and deexcitation of heavy clusters by GEMINI. A
larger $\Delta r_{\mathrm{np}}$ associated with a larger $L$ value increases
the isospin asymmetry of spectator matter and thus leads to a larger $N_n/N_p$,
especially in ultracentral collisions where the multiplicity of free nucleons
are free from the uncertainties of cluster formation and deexcitation. We have
further shown that the double ratio of $N_n/N_p$ in isobaric collision systems
or in collisions by isotopes helps to cancel the detecting efficiency for
protons. Effects from nuclear deformation and electromagnetic excitation are
studied, and they are found to be subdominant compared to the expected
sensitivity to $\Delta r_{\mathrm{np}}$.
