The story is known only through Hávamál, a piece of poetry from the poetic Edda, in which it takes up two stanzas. The relevant here is stanza 138:
Veit ek, at ek hekk
vindga meiði á
nætr allar níu,
ok gefinn Óðni,
sjalfr sjalfum mér,
á þeim meiði,
er manngi veit
hvers af rótum renn.
Hávamál, stanza 138, from Heimskringla.no
I know that I hung
upon a windy tree
for nine whole nights,
wounded with a spear
and given to Odin,
myself to myself for me;
on that tree
I knew nothing
of what kind of roots it came from.
The relevant word here is "gefinn". This appears to be a normal verb, the Old Norse equivalent of "given". So no, there was no specific term for this sacrifice, at least none that we know of.
Not sure if this is what you're looking for, but it looked promising.
"Odinn and His Cosmic Cross" (pdf) https://skemman.is/bitstream/1946/27093/1/Thesis%20Final%20Draft%20pdf.pdf
The term used is sjalsforn (self-sacrifice). "...sjálfsfórn contains metaphysical concepts found in religions across the globe..."
"This work will broaden the discussion surrounding sjálfsfórn by comparing it with various Indo-European myths and rituals. The oldest of these Indo-European traditions is the Rudra initiation, where the center pole of the initiation hut represents the center of the universe as well as the personification of Rudra. Shamanic initiations also display parallels to Óðinn’s self-hanging and this thesis will compare these rituals as well. Finnish shaman initiates must undergo torment to gain power, like Óðinn does in sjálfsfórn, and they must climb a tree, which represents the cosmic tree. The Sun Dance of the Plains Native Americans is a distant but similar ritual. Like Óðinn, the Sun Dance initiate is tethered to a tree, which represents the axis mundi, and sometimes they even hang from the tree. All of these myths and traditions convey the concept of the cosmic pillar as the ultimate liminal space, as well as the ideal location for sacrifice and initiation.
"To carry out these comparisons, this thesis is based upon Schjødt’s four levels of comparison. It compares sjálfsfórn with the rest of the Old Norse Corpus, with cultures neighboring Scandinavia, with other Indo-European cultures, and with typologically similar rituals such as the Sun Dance. This methodology will serve to reveal the self-hanging’s relationships to other myths and rituals, and it will highlight sjálfsfórn’s place in mythic tradition."