  By extending the methods used in our earlier work, in this paper, we present
an explicit realization of logarithmic $\mathcal{W}_{p,p'$}-modules that have
L(0) nilpotent rank three. This was achieved by combining the techniques
developed in \cite{AdM-2009} with the theory of local systems of vertex
operators \cite{LL}. In addition, we also construct a new type of extension of
$\mathcal{W}_{p,p'}$, denoted by $\mathcal{V}$. Our results confirm several
claims in the physics literature regarding the structure of projective covers
of certain irreducible representations in the principal block. This approach
can be applied to other models defined via a pair screenings.
