We report here an experimental approach that enables measurement of weak transitions to a wide range of rovibrational levels of water in the energy region 27000-34200 cm-1. We have previously demonstrated the use of laser double-resonance overtone excitation to access highly excited vibrational levels from single rovibrational states. While this approach simplifies the assignment of the spectra, it strongly reduces the number of observed transitions and hence our ability to test theoretical predictions. Here we increase significantly the number of observed transitions by allowing rotational and vibrational relaxation of H2O at intermediate levels of the double-resonance excitation scheme. Our recently developed semi- empirical potential energy surface PES12 (P. Maksyutenko, J. S. Muenter, N. F. Zobov, S. V. Shirin, O. L. Polyansky, T. R. Rizzo, and O. V. Boyarkin, J. Chem. Phys. 126, 241101 (2007)) enables assignment of the resulting complex spectra and reproduction of the measured transitions with sub-cm-1 accuracy.