Q: *How are two Helmholtz decompositions related?*

**A:** The scalar fields differ by a *harmonic function*.

Starting from a first decomposition $\sigma_1,\Gamma_1$, you can construct a second one by adding to $\sigma_1$ a harmonic function $h$, 
$$\sigma_2=\sigma_1+h,\;\;\text{with}\;\;\nabla^2 h=0.$$
Then the matching $\Gamma_2$ is determined by the equation
$$\nabla\times(\Gamma_1-\Gamma_2)=\nabla h.$$