Task-oriented handovers (TOH) are fundamental to effective human-robot collaboration, requiring robots to present objects in a way that supports the human’s intended post-handover use. Existing approaches are typically based on object- or task-specific affordances, but their ability to generalize to novel scenarios is limited. To address this gap, we present AFT-Handover, a framework that integrates large language model (LLM)-driven affordance reasoning with efficient texture-based affordance transfer to achieve zero-shot, generalizable TOH. Given a novel object-task pair, the method retrieves a proxy exemplar from a database, establishes part-level correspondences via LLM reasoning, and texturizes affordances for feature-based point cloud transfer. We evaluate AFT-Handover across diverse task-object pairs, showing improved handover success rates and stronger generalization compared to baselines. In a comparative user study, our framework is significantly preferred over the current state-of-the-art, effectively reducing human regrasping before tool use. Finally, we demonstrate TOH on legged manipulators, highlighting the potential of our framework for real-world robot-human handovers.