We introduce a strategy to study the parameter space of the general, CP-conserving, two-Higgs-doublet Model (2HDM) with a softly broken Z_2-symmetry by means of a new "hybrid" basis. In this basis the input parameters are the measured values of the mass of the observed Standard Model (SM)-like Higgs boson and its coupling strength to vector boson pairs, the mass of the second CP-even Higgs boson, the ratio of neutral Higgs vacuum expectation values, and three additional dimensionless parameters. Using the hybrid basis, we present numerical scans of the 2HDM parameter space where we survey available parameter regions and analyze model constraints. From these results, we define a number of benchmark scenarios that capture different aspects of non-standard Higgs phenomenology that are of interest for future LHC Higgs searches.