My brief researches only bring up the word "auntlike" to render the feminine equivalent of avuncular. Surely, though, and given the etymology of "aunt" [ < Latin amita -father's sister, old feminine past participle of amāre to love, i.e., beloved ], there must be a more lyrical word to hand.
Deriving from your own explanation in the OP, the natural choice would be amicular.
I do not seem to find any dictionary entries. Need to see why.
Book Doctor Gwen : 92 Feminine and Masculine Word Pairs
Feminine term / Masculine term /// neutral or inclusive term
4. amicular* / avuncular
(*Terms that are slang or recently coined.)
Contemporary Pragmatism - Google Books Result
John R. Shook, Paulo Ghiraldelli - 2004 - Philosophy - 200 pages
... be offered as amicular advice to discourse generation researchers, along the lines of the earlier 'Don't ask for the meaning; ask for the use', ...
The correct answer, courtesy of the Oxford English Dictionary, is the word materteral, whose entry I give here in toto:
Pronunciation: Brit. /məˈtəːtərl̩/ , U.S. /məˈtərdər(ə)l/
Etymology: < classical Latin mātertera maternal aunt ( < māter mother n.1 + ‑tera , feminine of ‑ter, suffix forming nouns) + ‑al suffix1.
Characteristic or typical of an aunt. Cf. avuncular adj.
- 1823 W. Taylor in Monthly Rev. 102 447 With maternal and materteral anxiety.
- 1867 J. N. Taylor Spindrift 6 You can picture the stately materteral form—A full-blown Atè, big with doom!
The proper citation for that entry is:
Third edition, March 2001; online version December 2011. < http://www.oed.com/view/Entry/114954 >; accessed 18 February 2012. An entry for this word was first included in New English Dictionary, 1905.
While their lyrical nature is subjective, dictionaries list both aunt-like as well as auntly as suitable adjectives.