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We study the secondary time-averaged flow (streaming) generated by an oscillating cylinder immersed within a fluid, under high amplitude forcing so that inertial effects are significant. This streaming is decomposed into a viscous boundary layer flow where vorticity is created, and an outer flow of larger size. We operate under conditions of relatively low viscosity, so that the boundary layer is smaller than the object diameter. While for low Keulegan-Carpenter (KC) number (small enough amplitude), the size of the outer flow is typically that of the object, here we show that at large enough forcing, the outer flow stretches along the direction of the vibration by up to 8 times, while the flow still keeps its axial symmetry. We quantify the elongation through PIV measurements under an unprecedented range of frequency and amplitude, so that the streaming Reynolds number reaches values much larger than unity. The absence of significant unsteady component of vorticity outside the viscous boundary layer - and the fact that the length of elongation scales well with the streaming Reynolds number - suggest that the stretching should be due to the convection of stationary vorticity by the streaming flow itself.