Все таки протомонгольское макалай, а литературное malaɣaj, "малгай" у халха-монголов.
Proto-Mongolian: *makalaj
Altaic etymology: Altaic etymology
Meaning: fur hat
Russian meaning: меховая шапка
Written Mongolian: malaɣaj (МХТТТ)
Middle Mongolian: maqalai, malaqaj (SH), maɣala (IM), maqalaj (MA)
Khalkha: malgaj
Buriat: malgaj
Kalmuck: maxlǟ
Dongxian: maGala (Tod. Dong.)
Dagur: malgaj (MD)
Monguor: marGa
Mogol: malGɛi (Weiers)
Comments: KW 254. Mong. > Man. maxala, see Doerfer MT 138, Rozycki 152-153.