It is not clear why you think Boltzmann's probability distribution should play a role in derivation of 2nd law. 2nd law states something about possible values of macroscopic constraints like $V,T$ at the end of processes that begin and end at some equilibrium state. Boltzmann's distribution depends on such variables as $V,T$, but does not place any constraint on them. So the relation is more likely the opposite: from the 2nd law of thermodynamics, or some equivalent (maximum information entropy for isolated system), one can derive the Boltzmann distribution.
If you want to derive 2nd law, this can be done in several ways, but one has to assume some equivalent of it. For example, there is the method of information entropy, introduced to thermodynamics by E.T. Jaynes. In this method, one assumes that thermodynamic entropy is maximum possible information entropy. See my answer here:
Understanding Gibbs $H$-theorem: where does Jaynes' "blurring" argument come from?