Strong gravitational lensing can produce copies of gravitational-wave signals
from the same source with the same waveform morphologies but different
amplitudes and arrival times. Some of these strongly-lensed gravitational-wave
signals can be demagnified and become sub-threshold. We present TESLA-X, an
enhanced approach to the original GstLAL-based TargetEd Subthreshold Lensing
seArch (TESLA) method, for improving the detection efficiency of these
potential sub-threshold lensed signals. TESLA-X utilizes lensed injections to
generate a targeted population model and a targeted template bank. We compare
the performance of a full template bank search, TESLA, and TESLA-X methods via
a simulation campaign, and demonstrate the performance of TESLA-X in recovering
lensed injections, particularly targeting a mock event. Our results show that
the TESLA-X method achieves a maximum of ∼20% higher search sensitivity
compared to the TESLA method within the sub-threshold regime, presenting a step
towards detecting the first lensed gravitational wave. TESLA-X will be employed
for the LIGO-Virgo-KAGRA's collaboration-wide analysis to search for lensing
signatures in the fourth observing run