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 27M⊙. 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.