  We show that certain type of tree forcings, including Sacks forcing,
increases the covering of the strong measure zero ideal $\mathcal{SN}$. As a
consequence, in Sacks model, such covering number is equal to the size of the
continuum, which indicates that this covering number is consistently larger
than any other classical cardinal invariant of the continuum. Even more, Sacks
forcing can be used to force that
$\mathrm{non}(\mathcal{SN})<\mathrm{cov}(\mathcal{SN})<\mathrm{cof}(\mathcal{SN})$,
which is the first consistency result where more than two cardinal invariants
associated with $\mathcal{SN}$ are pairwise different. Another consequence is
that $\mathcal{SN}\subseteq s^0$ in ZFC where $s^0$ denotes the Marczewski's
ideal.
