Subject to my previous question, I did some looking and found the relevant local data needed to construct a connection here.
To summarise the paper, given a connection $\nabla$ on a vector bundle $E$ whose local expression is $d + A^\alpha$ (for some cover of local trivialisations $\mathcal{U} = \{U_\alpha \}$), we can get an expression for $A^\alpha$ in terms of $A^\beta$ and the transition functions $g^\alpha_\beta$:
$$A^\alpha = (\textrm{d}g^\alpha_\beta)g^\beta_\alpha + g^\alpha_\beta A^\beta g^\beta_\alpha$$
In the paper it says that we can use a partition of unity to construct $\nabla$ from these local expressions, but I cannot seem to do this.
My guess would be something like this: suppose we have a partition of unity $\{\psi_\alpha\}$ subordinate to the cover $\mathcal{U}$, and some section $s \in \Gamma(E)$. Then we can define: $$ \nabla(s) = \sum_\alpha \psi_\alpha \nabla^\alpha(s|_{U_\alpha}) $$
Here I am just mimicing the construction I've seen in the proof that every vector bundle admits a connection.
I do not know how to proceed from here - how do I show that this is a connection? I know that I have to show that the Leibniz identity is satisfied, however it is unclear how I can do this, and moreover, how it relates to the transition data above.