That's pretty much what they're planning to do -- except that the planned target is a smaller Kuiper Belt Object, not a dwarf planet.
UPDATE : I overestimated the course change possible by using Pluto's gravity. See David Hammen's answer for the details.
After the Pluto flyby, the current plan is for New Horizons to continue on its path and fly by one or two Kuiper Belt objects (KBOs). Two such objects have been identified as possible targets. As far as I can tell a selection has not yet been made. One is estimated to be about 30-45 kilometers in diameter; the other is likely somewhat larger.
The choice of target is tightly constrained by available fuel for course adjustments. Before October 2014, there were no known bodies beyond the Pluto system that New Horizons would be able to reach. A search was conducted using ground telescopes, and later the Hubble Space Telescope.
Once a target is chosen, the flight path will be adjusted during the Pluto flyby to ensure that it passes near the selected object. Pluto's gravity will have to be taken into account when planning the course correction.
It's not clear that this would be a "slingshot". The goal is not necessarily to increase the spacecraft's velocity, as was done by the Jupiter flyby in 2006; rather it will select whatever course is needed to reach the target.
A flyby of a another dwarf planet after leaving the Pluto system is, unfortunately, not feasible. The choice of targets past Pluto is tightly constrained by fuel, distance, and time. The spacecraft has only a very limited amount of hydrazine propellant available for maneuvers, and it can't effectively return good science data past about 55 AU. This limits available targets to a cone less than 1 degree wide. There just aren't any dwarf planets past Pluto that New Horizons is able to reach. Only three objects past Pluto have been officially recognized as dwarf planets; a handful of others are likely.
This information is summarized from the Wikipedia article.
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