The Atiyah-Hirzebruch spectral sequence

\begin{equation*}E^2_{p,q}=H_p(B,h_q(F))\Rightarrow h_{p+q}(E),\end{equation*}

computes the generalized homology $h$ of a total space $E$ of a Serre fibration from the ordinary homology $H$ of the base space $B$ with coefficients in the generalized homology of the fiber $F$. In particular, using the fibration $\text{pt}\rightarrow X\rightarrow X$ computes $h_n(X)$ from $h_n(\text{pt})$ and knowledge of the differentials.

For simplicity, suppose that $h$ satisfies $h_0(\text{pt})=\mathbb{Z}$. (This may not be necessary in the end.)

As a generalized homology theory , $h$ corresponds to a topological spectrum (also called $h$). Trivially, the Eilenberg-Maclane spectrum $H$ is a Postnikov truncation of $h$; in particular, the homotopy groups agree at degree $0$ and there are no k-invariants to check.

My intuition is that the spectral sequence refines $H$ by incorporating data from the higher degrees of the Postnikov tower of $h$. The new homotopy groups $\pi_n(h)$ in the Postnikov tower appear as the coefficient modules $h_n(\text{pt})$ in the spectral sequence, while the new k-invariants appear as differentials.

Now, my question. Suppose $h$ and $g$ are topological spectra such that $g$ is a Postnikov truncation of $h$. Is there an analog to the Atiyah-Hirzebruch spectral sequence that computes $h$ from $g$?

(Perhaps Postnikov truncation is too strong or too weak an assumption. Please feel free to interpret my question to make it more interesting!)