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New Horizons will fly past the Kuiper Belt object 2014 MU69 in January 2019. This will use 56 m/s of the remaining ∆V budget of 130 m/s (1). The encounter will occur near the inner edge of the belt:
It looks perhaps feasible to find another target, but only if they look for one. Is it, and will they?
It's unlikely.
Per one of the scientists working navigation on the New Horizons mission, Simon Porter,
Lack of known objects prevents [targeting an additional flyby]; PT1 is almost on the outer edge of the [cold classical Kuiper Belt].
Many expensive hours of Hubble time and months of calendar time were dedicated to finding the three candidate objects. There is not sufficient delta V to visit more than one of the detected objects, so any possible third encounter would need to be an as-yet-undetected object significantly further (and fainter) than PT1. It's probably safe to assume that anything detectable by Hubble would already have been found, and we will have no assets that outperform Hubble operational prior to the PT1 encounter.
There's no particular reason why a new target has to be found before the PT1 encounter - in particular PT1 is far too small to provide a useful gravity assist, so a maneuver could be made some time after that to target a yet-to-be-discovered deeper object. Of course, the longer you wait, the smaller the region of reachable targets.
If JWST launches on time (haha), it is slated to complete commissioning a few months after the PT1 encounter. Its limiting magnitude will be approximately 1 magnitude fainter than Hubble for a comparable length of exposure. 1 magnitude is not a tremendous increase in sensitivity, and the corresponding probability of finding an object that both wasn't already detected by Hubble and is reachable with NH's dwindling delta V reserve after the PT1 flyby is small. It will probably not be considered worthwhile to spend the JWST time on that search.
I'm confident that if New Horizon's can reach another target, they will attempt to visit it. One of the reasons PT1 was selected as the target is that it was the closest object they could reach, leaving a reasonable amount of delta v, about half of what remained after flying by Pluto. I'm also confident they will try to find such potential targets. However, I suspect it will be difficult to find another candidate at all that could be reached. Even the selected candidate now was difficult to find, and I suspect it would be even more challenging to find another suitable target. Still, no doubt an attempt to find a candidate will be attempted, if it looks like there is remaining fuel.
Also note that likely the fuel will be further used to refine the trajectory to be at a very optimal location, which will allow better science of the one target, but further reduce the likelihood of another flyby.
Note from this article the following quote:
It also keeps open a possibility for New Horizons to continue on a second extended mission after the flyby, potentially making astronomical observations and probing the boundary between the solar system and interstellar space.
That seems to be the current plan, although I would still be willing to bet that an attempt will be made to visit another object, given the opportunity.
EDIT: Another target will be sought, possibly using Hubble but also using LORRI to try and find a potential target. It seems unlikely another target will be found, but they will still make the attempt.
