This is not true. For example for $S^1$-spectra there is the descent spectral sequence $$ H^s_{Nis}(X,\pi_tN) \Rightarrow \pi_{t-s}\mathrm{Map}(\Sigma^∞_+X,N)=\mathrm{Hom}(\Sigma^∞_+X, N[t-s])$$ and the statement you want would be like saying that the $E_2$-page is concentrated in the $t=0$ line.
In fact I do not think that $S_N=\pi_0\mathrm{Map}(\Sigma^∞_+-,N)$ is a Nisnevich sheaf at all. This would be like saying that if a complex of presheaves satisfies hyperdescent, then all of its homology presheaves are sheaves.
The statement becomes true when $N$ is in the heart of the t-structure. In fact in that case the aforementioned spectral sequence is concentrated in the $t=0$ line (although this is circular, since we use that statement to construct the spectral sequence). One way to prove it is to show that $\pi_*\mathrm{Map}(\Sigma^∞_+X,-)$ is an effaceable $\delta$-functor which for $*=0$ coincides with the global sections (by definition). The key lemma is the fact that if $I$ is an injective sheaf of abelian groups on a site, then $H\circ I$ (where $H$ is the Eilenberg-MacLane functor) is a hypersheaf of spectra over that site. This is just the classical fact that Čech cohomology of injective sheaves of abelian groups is trivial.