tale otherwise so utterly improbable

Because of past interactions regarding it, I’m going to let the answer below stand in its already revised state, but I’ve been feeling like this answer and answers and comments from others have been circling around what needed to be seen. Having moved from the 1971 compact version of the OED to the current version online, I find a much fuller account of the history of otherwise, listing two meanings as noun, principally in adverb phrases, five as adverb, and two as adjective, with many more examples.

The confusion over the meaning of otherwise in the sentence arises from confusion over its role. It is not an adjective of tale nor an adverb of improbable. Agreeing in part with Anton, but arriving at a different conclusion, I see tale is followed by an implied relative pronoun and verb to be omitted from an adjectival relative clause for stylistic reasons. In that clause, otherwise acts as an adverb of the unstated verb:

“I remembered also the nervous fever with which I had been seized just at the time that I dated my creation, and which would give an air of delirium to a tale [which is] otherwise so utterly improbable.”

Substituting one of the OED’s expressions of meaning B.1.a.— “In another way or ways; in a different manner; by other means; in other words; differently”—the sentence can be read:

“I remembered also the nervous fever with which I had been seized just at the time that I dated my creation, and which would give an air of delirium to a tale [which is] [by other means] so utterly improbable.”

I also agree with Apollyon's sense that already is part of the meaning particular to this sentence; perhaps in getting to Shelley's intention it can, interpretively, be added to, rather than replacing, our sense of otherwise:

“I remembered also the nervous fever with which I had been seized just at the time that I dated my creation, and which would give an air of delirium to a tale [which is] [by other means] [already] so utterly improbable.”

As stated below, "air of delirium" strengthens the quality of unbelievability in the tale expressed by "so utterly improbable," such that the speaker determines to remain silent rather than try to take responsibility for the genesis of the monster.

Mary Shelley’s choosing to let the relative pronoun be implied is a clear stylistic improvement over using either which for the third time or that for the second time in the sentence.

((((The two meanings of "otherwise" are (my paraphrasing):

1. in a different manner, or by other means; in another way also coming to and strengthening or reaffirming said certain condition

2. on the other hand, being in a different condition

The usage I have paraphrased here in the first spot comes first in the OED and is probably the earlier one. Both senses inhere in the word. It is actually a phrase, "in other way," compressed to a word just as "for a while" has become "awhile." The loss of "in" at the beginning accompanies the loss of its original meaning. The first seems much less common in the record, but the phrase originates in Old English, so there are centuries to explain the progression of meaning prior to Middle and modern English. In recent times the first sense has been largely lost.

Examples of usage 1:

Edmund Gosse. The Life and Letters of John Donne, Dodd, Mead 1899; Peter Smith 1959 (Volume 1) p.196. Kindle Edition. Gosse dates this letter to 1608:

"'A gent, that visited me yesterday told me that our Church hath lost Mr. Hugh Broughton, who is gone to the Roman side. I have known before that Serarius the Jesuit was an instrument from Cardinal Baronius to draw him to Rome to accept a stipend, only to serve the Christian Churches in controversies with the Jews, without endangering himself to change of his persuasion in particular deductions between these Christian Churches, or being inquired of, or tempted thereunto. And I hope he is no otherwise departed from us.'"

Radcliffe, Ann. The Mysteries of Udolpho: A Romance, 1794. (Penguin Classics) (p. 74). Start Classics. Kindle Edition:

“The grandeur and sublimity of the scenes, amidst which they had first met, had fascinated her fancy, and had imperceptibly contributed to render Valancourt more interesting by seeming to communicate to him somewhat of their own character. The esteem, too, which St. Aubert had repeatedly expressed for him, sanctioned this kindness; but, though his countenance and manner had continually expressed his admiration of her, he had not otherwise declared it; and even the hope of seeing him again was so distant, that she was scarcely conscious of it, still less that it influenced her conduct on this occasion.”

Robert Browning, “Guido,” lines 644-648 in The Ring and the Book p 120. Smith, Elder and Co., London, 1869. At Internet Archive:

    "Straight he leaves lobby, trundles, two and two, 
    Down steps, to reach home, write if but a word  
    Shall end the impudence: he leaves who likes 
    Go pacify the Pope: there's Christ to serve!  
    How otherwise would men display their zeal?"

James J. O'Donnell, The Ruin of the Roman Empire, 2008. p315: Speaking of Patrick: "The Ireland that claimed him was otherwise little touched by the tendrils of Mediterranean civilization ...."

Examples of usage 2 are everywhere. Fielding, for example, uses otherwise forty-four times in Tom Jones, 1749, always in the second sense. It is often found in dictionaries to the exclusion of the first. However, one contemporary example of the first usage is noted.

Looking at the extended quotation of the paragraph offered by Xanne, the two elements linked by otherwise, "air of delirium" and "tale ... utterly improbable," converge in the same effect, that the speaker determines to remain silent. To me it seems usage 1 is meant in the passage from Mary Shelley's 1818 text. ))))