As suggested by Denis Nardin I am moving my comment here. However I don't know details well enough, so I am making this cw in case somebody can fill them in.
So, choose a cohomology theory $h^*$ like e. g. $K$-theory, and, given a cover $(U_i)_{i\in I}$ of $X$, let $p:Y\to X$ be the canonical map $\coprod_{i\in I}U_i\to X$. Then consider the associated simplicial space $$ \check C(p):=\left(Y\begin{smallmatrix}\leftarrow\\\rightarrow\\\leftarrow\end{smallmatrix} Y\times_XY\begin{smallmatrix}\leftarrow\\\rightarrow\\\leftarrow\\\rightarrow\\\leftarrow\end{smallmatrix}Y\times_XY\times_XY\begin{smallmatrix}\leftarrow\\\vdots\\\leftarrow\end{smallmatrix}\cdots\right), $$ so that $\check C(p)_n=\coprod_{i_1,...,i_n}U_{i_1}\times_X\cdots\times_XU_{i_n}$.
A result of Dugger (found in "Topological Hypercovers") shows that geometric realization of this simplicial space is weakly homotopy equivalent to $X$, and so is the homotopy colimit of it. This induces isomorphisms on $h^*$ between the geometric realization of $\check C(p)$ to $X$, and between the homotopy colimit of $\check C(p)$ to $X$.
If $\check C(p)$ were Reedy cofibrant, Theorem 5.83, on page 163 of "Generalized Cohomology" by Kōno and Tamaki, would say there is a spectral sequence converging (again under some mild conditions) to $h^*$ of the geometric realization, with the second page given by cohomology of the cochain complex corresponding to the cosimplicial abelian group $h^*(\check C(p))$. This would be the Moore normalization of the latter complex (given by restricting to nondegenerate simplices of $\check C(p)$) has $\prod_{\{i_1,...,i_n\}\subseteq I}h^*(U_{i_1}\cap\cdots\cap U_{i_n})$ in the $n$th degree, with understandable differentials.
But $\check C(p)$ is not in general Reedy cofibrant. One way to recover the result would be to take a cofibrant replacement of $\check C(p)$ first and then take the cohomology spectral sequence of that resulting simplicial space. This is precisely the homotopy colimit spectral sequence, but unfortunately does not have as nice a description at the second page; it involves taking the homology of the Moore normalization of the simplicial replacement.
This generalizes (Dugger's Primer, Prop. 18.17): given a spectrum $\mathscr E$, and a diagram $F : \mathbf C \to \textbf{Top}$, there is a spectral sequence describd by taking the left derived colimit $$ E_{s, t}^2 = H_s(\mathbf C, \mathscr E_t(F)) = \operatorname{colim}_s E_t(F) \Rightarrow E_{s + t}(\operatorname{hocolim}_{\mathbf C}(F)) \cong E_{s + t}(X). $$
Here, we would choose the diagram to be the above simplicial space, and the desired K-Theory spectrum.
