Let $L\to M$ be a topologically trivial complex Hermitian line bundle (over a manifold of dimension three, if this is of any importance). I assume that $L$ admits a trivialization, however, I do not assume that any particular trivialization is chosen.
The gauge group $\mathcal G:=C^0(M; U(1))$ acts on the space of trivializations in a natural way and this action is transitive. It is easy to see that $\mathcal G$ is not connected in general. In fact, we have a natural homomorphism $\mathcal G\to H^1(M;\mathbb Z), f\mapsto f^*a$, where $a$ is a generator of $H^1(U(1);\mathbb Z)$. This in turn yields an action of $H^1(M;\mathbb Z)$ on the space $C$ of connected components of the space of all trivializations of $L$, i.e., $C$ is an $H^1(M;\mathbb Z)$-torsor. So, the question is whether there is a natural choice of the origin in $C$?