I find that in modern books on differential topology, the definition of fibre bundles is often given in terms of a fibre bundle charts. In other words, we have the following definition:
A fibre bundle is a quadruplet $(E,M,\pi;F)$ where $E$, $M$, and $F$ are smooth manifolds, and $\pi$ is a smooth surjective map $E\rightarrow M$ such that there exists a covering of $M$ by "fibre bundle charts" $(U_i,\phi_i)_{i\in I}$ where each $\phi_i:\pi^{-1}(U_i)\rightarrow U_i\times F$ is a diffeomorphism satisfying: $$\pi_{U_i}\circ \phi_i=\pi$$
However, in most older textbooks from the 20th century the definition of a fibre bundle is often given in terms of transition functions which satisfy the cocycle conditions. Now I understand how to move from the definition above to obtain transition functions which satisfy the cocycle conditions, but what I don't understand is how to obtain an atlas of bundle charts from the transition functions.
In other words, given transition functions $\phi_{ij}:U_i\cap U_j\rightarrow \text{Diff}(F)$ which satisfy the cocycle conditions, how do I obtain a covering of $M$ by fibre bundle charts?