We could easily overtake Voyager via only Jupiter if we wanted to (and New Horizons eventually will), and Jupiter-to-any-target launch windows come at least every 12 years.
https://www.universetoday.com/154791/the-best-way-to-leave-t...
(I have also seen a variant with an upper stage class SRB used for the meneuver.)
"The beauty of the gravity assist is that you use the gravity field of a large body to change course. A common misconception is that the gravity assist increases speed, but it actually leaves speed unchanged. It's more accurate to say that the gravity assist changes direction, since velocity is both a magnitude (speed) AND a direction."
And that's assuming you don't do an extra burn at periapsis, which is far more efficient at changing speed than doing the burn in interplanetary space.
Wiki explains it well. From the frame of reference of ship and planet, the relative speed is the same. But relative to the sun, the ship can be moving faster.
"A gravity assist around a planet changes a spacecraft's velocity (relative to the Sun) by entering and leaving the gravitational sphere of influence of a planet. The sum of the kinetic energies of both bodies remains constant (see elastic collision). A slingshot maneuver can therefore be used to change the spaceship's trajectory and speed relative to the Sun"