  We propose a new strategy of finding strongly-lensed supernovae (SNe) by
monitoring known galaxy-scale strong-lens systems. Strongly lensed SNe are
potentially powerful tools for the study of cosmology, galaxy evolution, and
stellar populations, but they are extremely rare. By targeting known strongly
lensed starforming galaxies, our strategy significantly boosts the detection
efficiency for lensed SNe compared to a blind search. As a reference sample, we
compile the 128 galaxy-galaxy strong-lens systems from the Sloan Lens ACS
Survey (SLACS), the SLACS for the Masses Survey, and the Baryon Oscillation
Spectroscopic Survey Emission-Line Lens Survey. Within this sample, we estimate
the rates of strongly-lensed Type Ia SN (SNIa) and core-collapse SN (CCSN) to
be $1.23 \pm 0.12$ and $10.4 \pm 1.1$ events per year, respectively. The lensed
SN images are expected to be widely separated with a median separation of 2
arcsec. Assuming a conservative fiducial lensing magnification factor of 5 for
the most highly magnified SN image, we forecast that a monitoring program with
a single-visit depth of 24.7 mag (5$\sigma$ point source, $r$ band) and a
cadence of 5 days can detect 0.49 strongly-lensed SNIa event and 2.1
strongly-lensed CCSN events per year within this sample. Our proposed
targeted-search strategy is particularly useful for prompt and efficient
identifications and follow-up observations of strongly-lensed SN candidates. It
also allows telescopes with small field of views and limited time to
efficiently discover strongly-lensed SNe with a pencil-beam scanning strategy.
