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 star-forming 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 +/- 0.12 and 10.4 +/- 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 fields of view and limited time to efficiently discover strongly lensed SNe with a pencil-beam scanning strategy.