Let $\{U_\alpha\}$ be a covering of a smooth manifold $M$. Replacing it by a refined covering if necessary, we may assume some good properties of it, like, (1) any intersection $\cap_{i=1}^k U_{\alpha_i}$ is contractible; or (2) it is locally-finite

Then, can we find a triangulation $\{\Delta_i\}$ of $M$ satisfying the following: for each $i$ there exists a $U_\alpha$ such that whenever $\Delta_j\cap \Delta_i \neq \varnothing$, we have $\Delta_j\subset U_\alpha$?

If this is not true, can we add some further conditions to make it true? If this is true, can you give some hints of proof or some reference? (I guess this should be true at least for compact $M$) Thanks!