Altermagnets host chirally split magnons that promise unique functionalities for information processing. However, their distinctive transport signatures, crucial for experimental identification and manipulation, remain elusive. Here, we predict that a spin accumulation electrically injects a ``vector" or multidirectional magnon spin current into an altermagnet, comprising both longitudinal and sizable transverse components. Notably, this transverse current exhibits a sign reversal away from the source and can be switched on or off by reorienting the N'eel vector. While such a transverse current is found to be not forbidden even in conventional antiferromagnets, we demonstrate through quantum-kinetic calculations that in altermagnets, the transverse response is enhanced by two orders of magnitude due to broken parity-time-translation ($PT\tau$) symmetry. This giant enhancement provides a decisive transport fingerprint for detecting magnon spin splitting and N'eel-vector orientation, offering a clear criterion to experimentally distinguish non-relativistic spin splitting from relativistic anisotropy effects.