New Horizons used its monopropellant attitude control thrusters for course correction. Because it couldn't make large course changes with the limited fuel available, the choice of targets was severely constrained:
Mission planners searched for one or more additional Kuiper belt objects (KBOs) of the order of 50–100 km (31–62 mi) in diameter as targets for flybys similar to the spacecraft's Plutonian encounter. However, despite the large population of KBOs, many factors limited the number of possible targets. Because the flight path was determined by the Pluto flyby, and the probe only had 33 kilograms of hydrazine propellant remaining, the object to be visited needed to be within a cone of less than a degree's width extending from Pluto. The target also needed to be within 55 AU, because beyond 55 AU, the communications link will become too weak, and the RTG power output will have decayed significantly enough to hinder observations.
It's also not making "many flybys" of Ultima Thule; it got one pass at Pluto in 2015 and one pass of Ultima Thule in 2018, and it is unclear if it will be able to steer toward a third target in the future.
This is typical of deep space probes. Flyby probes will use small attitude control thrusters, typically, since there's plenty of time to make course changes. For interplanetary orbiters, somewhat larger engines are generally needed for orbital insertion, and that usually makes it economical to use bipropellant thrusters (typically MMH/NTO or similar hypergolics) with better specific impulse than monoprops.
In contrast, a few interplanetary probes have used ion thrusters, which provide about 10x the specific impulse (fuel efficiency) of chemical rockets, albeit with extremely low thrust. The Dawn mission, for instance, was able to orbit two different dwarf planets in the asteroid belt, which is an unprecedented achievement.