Let $\mathbf{X}$ be a Grothendieck topos and let $A$ be an abelian group in $\mathbf{X}$. Verdier's Theorem allows one to describe $\mathrm{H}^n(\mathbf{X},A)$ in terms of hypercoverings, namely, as the colimit of $\mathrm{H}^n(U_\bullet,A)$ where $U_\bullet$ ranges over the category of hypercoverings (of the terminal object of $\mathbf{X}$).

One can further form the Cech cohomology $\check{\mathrm{H}}^n(\mathbf{X},A)$, which is the colomit of $\mathrm{H}^n(U_\bullet,A)$ as $U_\bullet$ ranges over hypercoverings with $U_\bullet=\mathrm{cosk}_0(U_\bullet)$. Let me call them *Cech hypercoverings*. (These are the hypercoverings with $U_n=U_0^{\times (n+1)}$ and the standard simplicial structure.)

A well-known spectral sequence relates the derived-functor cohomology with the Cech cohomology, implying in particular that $\mathrm{H}^1(\mathbf{X},A)$ is canonically isomorphic to $\check{\mathrm{H}}^1(\mathbf{X},A)$. I am not absolutely certain, but it seems correct that under the description provided by Verdier's Theorem, the isomorphism $\check{\mathrm{H}}^1(\mathbf{X},A)\to \mathrm{H}^1(\mathbf{X},A)$ is the obvious one, namely, if a *Cech* cohomology class is represented by a $1$-cocycle in $Z^1(U_\bullet,A)$, then its image in $\mathrm{H}^1(\mathbf{X},A)$ is the cohomology class represented by that $1$-cocycle.

The latter statement means that for a hypercovering $U_\bullet$ (not necessarily Cech) and any $1$-cocycle $\alpha\in Z^1(U_\bullet,A)$, one can find a *Cech* hypercovering $U'_\bullet$ and $\alpha'\in Z^1(U'_\bullet,A)$ representing the same cohomology class.
**I am looking for a way to construct these $U'$ and $\alpha'$ directly.**
The main problem is that $U_\bullet$ cannot be refined to a Cech hypercovering in general.

An alternative approach to the problem (which also applies to non-abelian $A$) is via the correspondence with $A$-torsors. It is a standard fact that $\check{H}^1(\mathbf{X},A)$ is in 1-1 correspondence with isomorphism classes of $A$-torsors. Explicitly, if $\alpha\in Z^1(U_\bullet,A)$ with $U_\bullet$ being a Cech hypercovering, then the $A$-torsor $P$ corresponding to $\alpha$ can be described by $$ P(V)=\{a\in A(U_0\times V)~:~ \alpha_V \cdot d_0^1a =d_1^1 a\} $$ where $d^1_i:A(U_0\times V)\to A(U_1\times V)=A(U_0\times U_0\times V)$ is induced by $d^1_i:U_1\to U_0$. Suppose now that $U_\bullet$ is an arbitrary hypercovering (not necessarily Cech). The question will be resolved if one can show that $P$ constructed above is still an $A$-torsor. The difficult thing to check is that $P\neq \emptyset$. (However, when $U_\bullet$ is Cech, one can check that $\alpha\in P(U_0)$.)