A single-crystal sample of the frustrated quasi one-dimensional quantum magnet Cs$_{2}$Cu$_{2}$Mo$_{3}$O$_{12}$ is investigated by magnetic and thermodynamic measurements.A combination of specific heat and magnetic torque measurements maps out the entire $H$-$T$ phase diagram for three orientations.Remarkably, a new phase emerges below the saturation field, irrespective of the crystal orientation. It is suggested that the presaturation phase represents spin-nematic order or other multi-magnon condensate. The phase diagrams within the long-range ordered dome are qualitatively different for each geometry. In particular, multiple transitions are identified in the field along the chain direction.
