In this paper we give a structural characterization and extend the tractability frontier of the Simple Temporal Problem (STP) by defining the class of the Extended Simple Temporal Problem (ESTP), which augments STP with strict inequalities and monotone Boolean formulae on inequations (i.e., formulae involving the operations of conjunction, disjunction and parenthesization). A polynomial-time algorithm is provided to solve ESTP, faster than previous state-of-the-art algorithms for other extensions of STP that had been considered in the literature, all encompassed by ESTP. We show the practical competitiveness of our approach through a proof-of-concept implementation and an experimental evaluation involving also state-of-the-art SMT solvers.