The formation of a cold Jupiter (CJ) is expected to quench the influx of pebbles and the migration of cores interior to its orbit, thus limiting the efficiency of rocky planet formation either by pebble accretion and/or orbital migration. Observations, however, show that the presence of outer CJs (>1 au and greater than or similar to 0.3M(Jup)) correlates with the presence of inner super-Earths (at <1 au). This observation may simply be a result of an enhanced initial reservoir of solids in the nebula required to form a CJ or a yet-to-be-determined mechanism assisted by the presence of the CJ. In this work, we focus on the latter alternative and study the orbital transport of planetesimals interior to a slightly eccentric (similar to 0.05) CJ subject to the gravity and drag from a viscously evolving gaseous disk. We find that a secular resonance sweeping inward through the disk gradually transports rings of planetesimals when their drag-assisted orbital decay is faster than the speed of the resonance scanning. This snowplow-like process leads to large concentration (boosted by a factor of similar to 10-100) of size-segregated planetesimal rings with aligned apsidal lines, making their expected collisions less destructive, due to their reduced velocity dispersion. This process is efficient for a wide range of alpha-disk models (and thus disk lifetimes) and Jovian masses, peaking for values similar to 1-5M(Jup), which are typical of observed CJs in radial velocity surveys. Overall, our work highlights the major role that the disk's gravity may have on the orbital redistribution of planetesimals, depicting a novel avenue by which CJs may enhance the formation of inner planetary systems, including super-Earths and perhaps even warm and hot Jupiters.