A homological condition that might be useful: in the Hopf case, the Yoneda algebra
$Ext_A^\bullet(k,k)$ embeds into the Hochschild cohomology
$HH^\bullet(A,A)$, HH^\bullet(A,A)$, moreover, there is a Gerstenhaber algebra structure on the Yoneda algebra, and this embedding is an embedding of Gerstenhaber algebras.
Reference: this article of Marco Farinati and Andrea Solotar.
I have a feeling that it would give some information already for exterior algebras, though I don't have time to check it carefully now. Of course, to use this observation for exterior algebras, the graded commutative product from the Gerstenhaber structure (highlighted by mt in his answer) is enough. But I think that there are cases where the Lie bracket will help to settle the answer.