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.