I know little of the topic but the following may still be useful:
Burt Totaro, an expert, posts here about why we should believe the Hodge conjecture. See also the comments.
James Milne discusses absolute Hodge classes in relation to the Tate conjecture here. In particular he remarks that the Hodge conjecture follows from the Tate conjecture+Deligne's conjecture (essentially that all Hodge classes are absolute). He also mentions André's concept of motivated cycles on algebraic varieties, those are (images under correspondences of) cycles generated by applying Lefschetz operators to algebraic cycles. André proved that all Hodge classes are motivated, for abelian varieties. I think if the Lefschetz operators are algebraic André's motivated cycles are absolute Hodge, but I am not sure.
In the Charles-Schnell review of absolute Hodge classes (c.f. Parsa' comment), they mention Voisin's result that a general (Kähler) nonalgebraic Weil torus has nontorsion Hodge classes but trivial (or at least torsion) corresponding algebraic cohomology (algebraic classes). They remark this indicates that algebraicity of the manifold is probably a natural requirement, as there is no natural generalization of the Hodge conjecture to nonalgebraic Kähler manifolds.
Non-simply connected Calabi-Yau varieties generalize abelian varieties so this may be a class of varieties of interest for you.
I think there are few results extending Deligne's on absolute Hodge classes on abelian varieties. I am not aware of any. There may be more regarding André's motivated cycles.
Finally a result related to HC for abelian varieties is that mentioned by Totaro, that Tate proved the Tate conjecture for divisors on abelian varieties, which together with Deligne's result on absolute Hodge classes implies the Hodge conjecture for divisors on abelian varieties (which was known much before, in greater generality).
EDIT: Charles and Schnell give results on the absoluteness and the Lefschetz isomorphism. I would have to think how that relates to motivated cycles, looking at André's proof that Hodge classes are motivated.
PS: Charles and Schnell say Deligne's notion of absolute Hodge class is among the first appearances of the notion of motive but I wonder if this is not a little misleading. It seems to me there were many before, even in the work of Weil, leading to the Weil conjectures, or Serre's proof of them on the basis of the standard conjectures. I would have thought that Deligne provided justification for those ideas, more than bringing forth the intuition. But I am usually wildly mistaken on such things so do not trust my feeling much.