This paper presents several versions of the Ancona inequality for finitely supported, irreducible random walks on non-amenable groups. We first study a class of Morse subsets with narrow points and prove the Ancona inequality around these points in any finitely generated non-amenable group. This result implies the inequality along all Morse geodesics and recovers the known case for relatively hyperbolic groups. We then consider any geometric action of a non-amenable group with contracting elements. For such groups, we construct a class of generic geodesic rays, termed proportionally contracting rays, and establish the Ancona inequality along a sequence of good points. This leads to an embedding of a full-measure subset of the horofunction boundary into the minimal Martin boundary. A stronger Ancona inequality is established for groups acting geometrically on an irreducible CAT(0) cube complex with a Morse hyperplane. In this setting, we show that the orbital maps extend continuously to a partial boundary map from a full-measure subset of the Roller boundary into the minimal Martin boundary. Finally, we provide explicit examples, including right-angled Coxeter groups (RACGs) defined by an irreducible graph with at least one vertex not belonging to any induced 4-cycle.