Let G be a discrete or say for sake of simplicity a finite group. In Hatcher's book Algebraic Topology on p 89 the construction of universal bundle $EG$ carries structure of a $\Delta$-complex whose $n$-simplices are the ordered $(n + 1)$ tuples $[g_0, ... ,g_n]$ of elements of $G$. As quotient space the model of the classifing space $BG=EG/G$ inherits structure $\Delta$-complex where a $n$-simplex $BG$ can be written uniquely in the form $[g_1 \vert g_2\vert ... \vert g_n]:= G \cdot [e_G, g_1, g_1g_2,..., g_1,.., g_n]$. Note that by construction $EG$ is realized even as a simplicial complex, while $BG$ inherits only structure of honest $\Delta$-complex due to more subtle identificiations by passing to quotient.

My question is if there is a rather "canonical" way to choose a trivializing cover $\{U_i\}_{i \in I}$ of the universal principal $G$-bundle $EG \to BG$, ie a family of open subsets of $BG$ such that $EG \vert _{U_i} \cong U_i \times G$ with relatively good controllable associated transition functions $h_{ij}: U_i \cap U_j \to G$ dictating the patching structure $ U_i \cap U_j \times G \to U_i \cap U_j \times G, (u,g) \mapsto (u, h_{ij}(u)g)$.

The question is closely cross posted from MSE where it not received any resonance.