Bertrand Russell as Madame Sosostris
Part I
Lines 43-59

Madame Sosostris, famous clairvoyante,
Had a bad cold, nevertheless
Is known to be the wisest woman in Europe,
With a wicked pack of cards. ...
[ ... ]
Thank you. If you see dear Mrs. Equitone,
Tell her I bring the horoscope myself:
One must be so careful these days.

The odd phrasing at the start of this section bothered me for years:

Madame Sosostris, famous clairvoyante, had a bad cold, nevertheless is known to be the wisest woman in Europe, with a wicked pack of cards.

Let me paraphrase Eliot with:

Sosostris had a bad cold, nevertheless is known to be wise.

Having a bad cold usually is a temporary state while being "known to be wise" is an ongoing state; Sosostris was wise, is wise and likely will remain wise. The two states (having a cold and being wise) are tied together with the use of "nevertheless" and it eluded me how this cold would affect Sosostris's reputation. It might temporarily affect her abilities but why would it last long enough to change her ongoing reputation? Well, if we change my paraphrasing to:

Sosostris had a mental deficiency, nevertheless is known to be wise.

Sosostris had (and has) a mental defientcy, nevertheless is known to be wise.

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Those who have checked their copy of B.C. Southam's A Guide to the Selected Poems of T.S. Eliot know about the possible derivation of Madame Sosostris' name from Madame Sosestris in Aldous Huxley's Crome Yellow. In the novel a fair is thrown and the Mr. Scogan character volunteers to be a fortune teller. He dresses as a gypsy woman and takes the name Sesostris, the Sorceress of Ecbatana. (Scogan is generally seen to be a not very sympathic portrayal of Bertrand Russell, a portrayal seen and noted by Eliot.) The first half of chapter 27 of Crome Yellow describes Scogan's stint as Sesostris.

Eliot, prior to the publication of The Waste Land, had a reputation as a satirist. Russell had a reputation as a satyr. It should be noted that the title character of an earlier Eliot poem, Mr. Apollinax, (from Prufrock and Other Observations - 1917) is considered a portrait of Russell.

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Back to The Waste Land -- I believe that the lines at the start and end of Eliot's Sosostris passage are an extension of Huxley's satirizing of Russell. Sosostris is Sesostris is Scogan is Russell.

This leads to an explanation for Sosostris's bad cold. A visitor to our famous clairvoyante might mention how deep "her" voice sounds. A likely response from "Madame" Sosostris is that she is suffering from a bad cold, one that would be a permanent condition in her case.

As for Sosostris being "the wisest woman in Europe" this would refer to Russell's reputation for brilliance.

"With a wicked pack of cards" has a connotation of being evil because of an ungodly, occult aspect of the Tarot. Another, playful, use of "wicked" is the American slang connotation of being marvellous. But I propose that Eliot is also extending Huxley's protrayal of Scogan/Russell by suggesting that Sosostris is wicked and has a wicked use for the cards, that of seduction. This plays off the scene in Crome Yellow where Madame Sosestris slyly sets up a rendezvous with a young female client for the purpose of testing the woman's virtue.

Scogan is only playing a fortune teller as Sosestris. Since Eliot's Sosostris is an extension of Huxley's Sosestris the reason that she cannot find the Hanged Man or see what the one-eyed merchant carries is not because she is ill but because she/he is a fraud and a cheat. Just as Russell might know details of Eliot's life but not know the inner man, Sosostris can show her cards but cannot give them meaning.

The sexual wickedness of Scogan/Russell reappears at the end of the Sosostris section of The Waste Land. "Madame" Sosostris will be paying a visit to Mrs. Equitone. I suspect that they will not be discussing the horoscope. With a visit of this type the cautionary statement "one must be so careful these days" makes sense. Mrs. Equitone is a married woman after all.

I should mention that I still stand by my interpretation despite this comment by Southam:

Eliot said that he read the novel [Crome Yellow] on its publication in November 1921 and that it is 'almost certain' that he borrowed the name from it, although he was 'unconscious of the borrowing'.

Some additional speculations on Sosostris/Russell and Mrs. Equitone can be found on another commentary page.