The VEI 4 rhyolitic eruption of Cordón Caulle volcano in 2011-2012 was immediately followed by ∼0.77 m of exponentially decaying uplift during 2012-2015. In this study, we present evidence of additional transient pulses of inflation with interferometric synthetic aperture radar (InSAR) time series during 2016-2018. We also assess whether the 2012-2015 uplift can be explained by viscoelastic relaxation or not. InSAR detected ∼12 cm of uplift during 2016-2017 and ∼5 cm during 2017-2018. The three inflation episodes have very similar spatial scales and can be modeled with the same inflating sill (z ∼ 6 km), but their time evolution is significantly different. Numerical models of a pressurized reservoir surrounded by a viscoelastic shell do not have a better fit than a magma injection model to the 2012-2015 InSAR time series, indicating that magma injection is the most likely mechanism to explain this uplift signal. The spatial similarities of the three pulses suggest that they are produced by episodic magma injection. These magma injection pulses provide the heat to remobilize the crystal mush beneath the volcano on timescales of a few months, but the end of the uplift is not predicted by existing models. None of these uplift pulses were related to abnormal seismicity, and we speculate that they are mostly aseismic because they caused stresses of lower magnitude than the coeruptive stresses. The meter scale displacement observed at Cordón Caulle between 2007 and 2018 suggests that the volcano undergoes episodic cycles of inflation like those observed in silicic calderas.