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We report results of our low-temperature magneto electric transport measurements on micron-sized short cylinders of odd-parity, spin-triplet superconductor Sr$_2$RuO$_4$ with the cylinder axis along the $c$ axis. The in-plane magnetic field and measurement current dependent magnetoresistance oscillations were found to feature an amplitude much larger than that expected from the conventional Little-Parks effect, suggesting that the magnetoresistance oscillations originate from vortex crossing. The free-energy barrier that controls the vortex crossing was modulated by the magnetic flux enclosed in the cylinder, the in-plane field, measurement current, and structural factors. Distinct features on magnetoresistance peaks were found, which we argue to be related to the emergence of half-flux quantum states, but only in samples for which the vortex crossing is confined at specific parts of the sample.