Oberth effect is one:

https://en.m.wikipedia.org/wiki/Oberth_effect

Fire a super high thrust engine during flyby.

If the PBH were in orbit around the Sun I don’t see why a conventional gravity assist would not work, but an Oberth effect maneuver would be more powerful.

When you have a conventional gravity assist, the speed (magnitude) is unchanged in the coordinate frame of the planet that's providing the assist. All that happens is the velocity vector is rotated in that frame. Thus the usefulness hinges on arriving with a large relative velocity, to start with—a large vector to rotate, allows for a large velocity change.

There's no heliocentric velocity in a slow-moving outer planet.