Indeed, as John Klein shows, the map is $(2k+2)$-connected. Let me offer an alternative proof of the fact that, for $X$ a $k$-connective spectrum, $k\geq 0$, the homomorphism $\pi_i\Sigma^\infty\Omega^\infty X\rightarrow\pi_i X$ is an isomorphism for $i\leq 2k+1$. As indicated by John, this homomorphism is surjective for all $i$ and it suffices to show that the canonical homomorphism $\pi_i\Omega^\infty X\rightarrow \pi_i\Sigma^\infty\Omega^\infty X$ is an iso for $i\leq 2k+1$. We will actually prove that this is more generally true replacing $\Omega^\infty X$ with a $k$-connected space $Y$ with abelian fundamental group and vanishing Whitehead products, e.g. a single loop space. I though that someone might eventually find this useful, even if it is not what you're asking about.

The Freudenthal suspension theorem shows that, for an $k$-connected space $Y$, the suspension operator
\[
\Sigma_*\colon \pi_i Y \longrightarrow \pi_{i+1}\Sigma Y
\]
is an isomorphism for $i\leq 2k$ and an epimorphism for $i=2k+1$. This and the fact that $\Sigma^j Y$ is $(j+k)$-connected clearly shows that the natural map from the homotopy groups of $Y$ to those of its suspension spectrum $\Sigma^\infty Y$
\[
\pi_i Y \longrightarrow \pi_{i}\Sigma^\infty Y,
\]
which is a transfinite composition of suspension operators, is an iso for $i\leq 2k$ and epi for $i=2k+1$.

Moreover, by the theorems of Blakers and Massey on the homotopy groups of triads, we have an exact sequence in the critical dimension
\[
\pi_{k+1}Y\otimes \pi_{k+1}Y\stackrel{[-,-]}\longrightarrow \pi_{2k+1}Y\stackrel{\Sigma_{*}}\twoheadrightarrow \pi_{2k+2}\Sigma Y
\]
where the first arrow is the Whitehead product. This still makes sense for $k=0$ using the non-abelian tensor product of groups and the Brown-Loday non-abelian Van Kampen theorems. In this case $[-,-]$ is the commutator product.

If Whitehad products in $\pi_*Y$ vanish and the fundamental group is abelian, then the suspension operator is also an isomorphism for $i=2k+1$, and we get an isomorphism between the homotopy groups of $Y$ and $\Sigma^\infty Y$ in this dimension too.

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