I think the point here is that this is a very long-period comet, so its speed is very close to solar system escape velocity right now. Even a tiny bit of extra energy from Jupiter can push its aphelion out a long way and significantly increase its period.
This is what Nick meant when he said that a small change in eccentricity has a big effect on period. When e is very close to 1 (i.e. a long-period comet), a very small change in e has a very big effect on the (1-e) term in Nick’s equation!