Layering of ocean velocity fine structure has been coherently observed across the entire extent of a Gulf Stream warm-core ring using a shipboard acoustic Doppler current profiler system in September 2009 and independently sampled as the ring transited a moored array. Lines of constant velocity phase generally followed isopycnals as they deepened within the ring center. We also observed a clear separation of the vertical structure of the flows associated with the ring (of order 0.5m/s) with the shorter (200m) and less energetic (similar to 0.2m/s) flows of the velocity fine structure, which was further observed to rotate clockwise with increasing depth, consistent with downward propagating near-inertial waves (NIWs). Observations are consistent with a ring-scale NIW packet, probably wind forced, that shows enhanced NIW energy within the sloping pycnocline at depths of 300-700m. Evidence of wind-forced NIWs within anticylonic eddies in a numerical simulation shows some similar features to our observations, which we try to understand physically with basic WKB-type wave/current dynamics along the lines of previously published work and a new calculation of NIW trapping within an isolated, baroclinic vortex.