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 $$\nabla\times(\Gamma_1-\Gamma_2)=\nabla h.$$