Recently, we have developed several techniques in order to accelerate Monte Carlo (MC) molecular simulations. For that purpose, two strategies were followed. In the first, new algorithms were proposed as a set of early rejection schemes performing faster than the conventional algorithm while preserving the accuracy of the method. On the other hand, a reweighting and reconstruction scheme was introduced that is capable of retrieving primary quantities and second derivative properties at several thermodynamic conditions from a single MC Markov chain. The latter scheme, was first developed to extrapolate quantities in NV T ensemble for struc- tureless Lennard-Jones particles. However, it is evident that for most real life applications the NpT ensemble is more convenient, as pressure and temperature are usually known. Therefore, in this paper we present an extension to the reweighting and reconstruction method to solve NpT problems utilizing the same Markov chains generated by the NV T ensemble simulations. Eventually, the new approach allows elegant switching between the two ensembles for several quantities at a wide range of neighboring thermodynamic conditions.