Yes, easily. The alignment doesn't really matter for that. Almost all your speed gain comes from just Jupiter. Saturn is 30% the mass and 2/3 of the orbital velocity, so your gain from Saturn is only 20% of what you can get from Jupiter (and also your potential gain is limited by a minimum approach distance greater than the rings, or you'd hit them.) And the ice giants are slower and smaller yet; Voyager barely gained from Uranus and actually slowed from Neptune since it wasn't routed to gain speed there.

New Horizons achieved 80% of Voyager's velocity with just Jupiter, and it wasn't really trying to optimize for speed, it approached Jupiter only to 10 million km (over 100x greater than the planet's radius.) A probe dedicated to a fast slingshot past Jupiter could easily overtake Voyager. We haven't had any need to try, unless one of the missions to specifically study the heliopause-interstellar area happens. It would still take a while to catch up to Voyager's head start, but it's doable.

The alignment for Voyager was captivating, but it really wasn't as important as people typically think. Jupiter alone can get you anywhere and launch windows for it come every 12 years. If the four-planet alignment hadn't happened then, realistically we would have just done separate Jupiter-Uranus and Jupiter-Neptune missions.