  In a projective plane $\Pi_{q}$ (not necessarily Desarguesian) of order $q$,
a point subset $\mathcal{S}$ is saturating (or dense) if any point of
$\Pi_{q}\setminus \mathcal{S}$ is collinear with two points in $\mathcal{S}$.
Modifying an approach of [31], we proved the following upper bound on the
smallest size $s(2,q)$ of a saturating set in $\Pi_{q}$: \begin{equation*}
s(2,q)\leq \sqrt{(q+1)\left(3\ln q+\ln\ln q
+\ln\frac{3}{4}\right)}+\sqrt{\frac{q}{3\ln q}}+3. \end{equation*} The bound
holds for all q, not necessarily large.
  By using inductive constructions, upper bounds on the smallest size of a
saturating set in the projective space $\mathrm{PG}(N,q)$ with even dimension
$N$ are obtained.
  All the results are also stated in terms of linear covering codes.
