I always assumed that the idea that Odin was hanged by the feet was diffused by new age interpretations related to the tarot card, but discussing this matter I have been told that norse animal sacrifices were hanged by the feet and then slaughtered, so Odin in "giving himself to himself" should have done the same
By the neck is much more likely (if this tradition really dates back to pre-Christian period).
Kure's seminal article comments on OP's question as following: "Though not specified by Hávamál, most readers take it for granted that Odin hung by his neck in the manner of an execution (Kure 2006: 69)."
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The following a small addition to her comment by me.
In the first stanza of Skaldic poem Háleygjatal (end of the 10th century) by Eyvindr skáldaspillir Finnsson that traces the linage of the patron, Jarl Håkon of Lade (d. 995) to the affair between Óðinn and Skaði calls Óðinn as falmr galga, "on the burden of the gallows" in one of its numerous kennings alluding to Óðinn himself:
"(Rough translation of the stanza of the poem) I would wish for a hearing for the drink of Hôarr <= Óðinn> [POETRY], while I lift up the payment for Gillingr <giant> [POETRY], while we [I] reckon his lineage back to the gods in the cauldron-liquid [DRINK] of the burden of the gallows [= Óðinn > POETRY], that which the travel-furtherer [= Óðinn] carried flying from the treasure-valleys of Surtr [giant] (Háleygjatal, Stanza 1: Translation is taken from the official site of Skaldic Poetry of the Scandinavian Middle Ages)."
Among other kennings (improvised compound with metaphorical interpretations) alluding to Óðinn (linked to some example, though evidently not so exhaustive, in the online database), this one is of especially importance in the following reasons:
So, unless the usual way of hanging the human (as an execution) in Viking Age Scandinavia was "by the feet.", it is natural to suppose that Poet Eyvindr imagine how Óðinn would have been hanged in accordance with the standard way of hanging, by the neck.
Kure also makes a brief summary of the research (interpretation) history of this famous episode in Stanzas. 138-41 of Hávamál, with two main trends, namely 1) sacrifice and 2) initiation, aside from the possible Christian influence. The second (initiation) hypothesis has got more support from researchers since 1970s (if they accept possible pre-Christian provenance of the tradition).
Concerning the correlation between the preferred way of sacrifice and the alleged symbolism of deities, as I also mentioned before in: Did Vikings sacrifice captives/enemies to gods and deities like the fire giant Sutr?, the majority of the scholars agree that is is difficult to see the exact correlation.
References:
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