Question

Some more data points from the Stacks Project:

smooth covers can be refined by etale covers (tag 055V)

tag 02LH could possibly also be useful, if we can identify what sort of map $\coprod_{i=1}^n T_i \to T \to S$ is:

Let $f : U \to S$ be a surjective etale morphism of affine schemes. There exists a surjective, finite locally free morphism $\pi : T \to S$ and a finite open covering $T = T_1 \cup \ldots \cup T_n$ such that each $T_i \to S$ factors through $U \to S$. if

Given an algebraic stack $X$ (defined as in Stacks Project), we can find a presentation by a smooth groupoid $R\rightrightarrows U$ in algebraic spaces (i.e. the source and target are smooth, and we have an equivalence $[U/R] \to X$). Tag 04T5 tells us that $X$ is also equivalent to the stack $[U'/R']$ where $R' \rightrightarrows U'$ is a presentation with source and target flat and locally of finite presentation (so there is a surjection which is flat and locally of finite presentation $U' \to X$).