The passage is just an extended metaphor. Mr. Martin is thinking that Ulgine Barrows had started her tenure as Mr. Fitweiler's special advisor by making minor organizational changes which were harmful to the firm's operations in small ways but had gathered confidence through the encouragement received from her employer and was now making major changes which, as Martin saw it, could destroy the entire organization. This kind of hyperbolical metaphor is not unusual in James Thurber's distinctive style of writing. He was a great admirer of Henry James and undoubtedly was strongly influenced by Henry James's so-called "later style," which is full of metaphors and similes. Here is an example from James's best novel The Ambassadors (1903). The protagonist, Lambert Strether, is delighted to find that he has some unexpected free time because his friend Mr. Waymarsh has not yet arrived, and James uses an extended simile in which he compares this unexpected reprieve from his friend's dour presence to finding money.
He winced a little, truly, at the thought that Waymarsh might be already at Chester; he reflected that, should he have to describe himself there as having "got in" so early, it would be difficult to make the interval look particularly eager; but he was like a man who, elatedly finding in his pocket more money than usual, handles it a while and idly and pleasantly chinks it before addressing himself to the business of spending.
The reader of Thurber's "The Catbird Seat" naturally understands that this reckless and incompetent Ulgine Barrows is "a loose cannon," a destructive force in the company to which Mr. Martin has dedicated so many years of loyal service. As an example of the damage Mrs. Barrows has already done, Martin reflects on how Mr. Fitweiler had met Mrs. Barrows at a party and "had jumped to the conclusion there and then that this was a woman of singular accomplishments, equipped to bring out the best in him and in the firm."
A week later he had introduced her into F & S as his special advisor. On that day confusion got its foot in the door. After Miss Tyson, Mr. Brundage, and Mr. Bartlett had been fired and Mr. Munson had taken his hat and stalked out, mailing in his resignation later, old Roberts had been emboldened to speak to Mr. Fitweiler. He mentioned that Mr. Munson's department had been "a little disrupted" and hadn't they perhaps better resume the old system there?
But Mr. Fitweiler, whom Thurber describes as "an aging gentleman," and who is seeing a psychiatrist, expressed his complete confidence in his new special advisor. With Fitweiler behind her, Ulgine Barrows could make any changes in any department. And as the story opens, it is evident that she is taking an interest in Erwin Martin's sacrosanct filing department.