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Strong gravitational lensing can produce copies of gravitational-wave signalsfrom the same source with the same waveform morphologies but differentamplitudes and arrival times. Some of these strongly-lensed gravitational-wavesignals can be demagnified and become sub-threshold. We present TESLA-X, anenhanced approach to the original GstLAL-based TargetEd Subthreshold LensingseArch (TESLA) method, for improving the detection efficiency of thesepotential sub-threshold lensed signals. TESLA-X utilizes lensed injections togenerate a targeted population model and a targeted template bank. We comparethe performance of a full template bank search, TESLA, and TESLA-X methods viaa simulation campaign, and demonstrate the performance of TESLA-X in recoveringlensed injections, particularly targeting a mock event. Our results show thatthe TESLA-X method achieves a maximum of $\sim 20\%$ higher search sensitivitycompared to the TESLA method within the sub-threshold regime, presenting a steptowards detecting the first lensed gravitational wave. TESLA-X will be employedfor the LIGO-Virgo-KAGRA's collaboration-wide analysis to search for lensingsignatures in the fourth observing run.

TESLA-X: An effective method to search for sub-threshold lensed gravitational waves with a targeted population model