We consider the numerical irreducible decomposition of a positive dimensional solution set of a polynomial system into irreducible factors. Path tracking techniques computing loops around singularities connect points on the same irreducible components. The computation of a linear trace for each factor certifies the decomposition. This factorization method exhibits a good practical performance on solution sets of relative high degrees. Using the same concepts of monodromy and linear trace, we present a new monodromy breakup algorithm. It shows a better performance than the old method which requires construction of permutations of witness points in order to break up the solution set. In contrast, the new algorithm assumes a finer approach allowing us to avoid tracking unnecessary homotopy paths. As we designed the serial algorithm keeping in mind distributed computing, an additional advantage is that its parallel version can be easily built. Synchronization issues resulted in a performance loss of the straightforward parallel version of the old algorithm. Our parallel implementation of the new approach bypasses these issues, therefore, exhibiting a better performance, especially on solution sets of larger degree.