A phrase/word for a Logical Jump in argument

There are a few terms for logical fallacies that might fit. In general discussion, I'd call it "jumping to conclusions".

hasty generalization

Hasty generalization is an informal fallacy of faulty generalization by reaching an inductive generalization based on insufficient evidence—essentially making a hasty conclusion without considering all of the variables.

faulty generalization

A faulty generalization is a conclusion about all or many instances of a phenomenon that has been reached on the basis of just one or just a few instances of that phenomenon. It is an example of jumping to conclusions.

proof by example

Proof by example (also known as inappropriate generalization) is a logical fallacy whereby one or more examples are claimed as "proof" for a more general statement.

You might also say that anecdotal evidence or cherry picking is being used.

Such a conclusion, which does not follow from the evidence or stated assumptions, is a non sequitur. Be a little careful though, as 'non sequitur' is (informally) also used to refer to statements which simply seem out of place.

A few things come to mind...

Fallacious: based on a mistaken belief.

Specious: superficially plausible, but actually wrong.

Ungrounded/unfounded: having no real basis or justification.

That being said, writing "the author made some leaps" or "the author was reaching with" seems acceptable.