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We prove smoothing properties of nonlocal transition semigroups associated to a class of stochastic differential equations (SDE) in ${ℝ}^d$ driven by additive pure-jump Lévy noise. In particular, we assume that the Lévy process driving the SDE is the sum of a subordinated Wiener process $Y$ (i.e. $Y=W\circ T$, where $T$ is an increasing pure-jump Lévy process starting at zero and independent of the Wiener process $W$) and of an arbitrary Lévy process independent of $Y$, that the drift coefficient is continuous (but not...