  We prove smoothing properties of nonlocal transition semigroups associated to
a class of stochastic differential equations (SDE) driven by additive pure-jump
L\'evy noise. In particular, we assume that the L\'evy process driving the SDE
is the sum of a subordinated Wiener process and of an independent arbitrary
L\'evy process, that the drift coefficient is continuous (but not necessarily
Lipschitz continuous) and grows not faster than a polynomial, and that the SDE
admits a Feller weak solution. By a combination of probabilistic and analytic
methods, we provide sufficient conditions for the Markovian semigroup
associated to the SDE to be strong Feller and to map $L_p$ to continuous
bounded functions. A key intermediate step is the study of regularizing
properties of the transition semigroup associated to the subordinated Wiener
process in terms of negative moments of the subordinator.
