  The gravitational wave (GW) and neutrino signals from core-collapse
supernovae (CCSNe) are expected to carry pronounced imprints of the standing
accretion shock instability (SASI). We investigate whether the correlation
between the SASI signatures in the GW and neutrino signals could be exploited
to enhance the detection efficiency of GWs. We rely on a benchmark full-scale
three-dimensional CCSN simulation with zero-age main sequence mass of $27\
M_\odot$. Two search strategies are explored: 1.~the inference of the SASI
frequency range and/or time window from the neutrino event rate detectable at
the IceCube Neutrino Observatory; 2.~the use of the neutrino event rate to
build a matched filter template. We find that incorporating information from
the SASI modulations of the IceCube neutrino event rate can increase the
detection efficiency compared to standard GW excess energy searches up to
$30\%$ for nearby CCSNe. However, we do not find significant improvements in
the overall GW detection efficiency for CCSNe more distant than $1.5$~kpc. We
demonstrate that the matched filter approach performs better than the unmodeled
search method, which relies on a frequency bandpass inferred from the neutrino
signal. The improved detection efficiency provided by our matched filter method
calls for additional work to outline the best strategy for the first GW
detection from CCSNe.
