We demonstrate propagation and small backscatter losses of a frequency-doubled (2omega) laser beam interacting with inertial confinement fusion hohlraum plasmas. The electron temperature of 3.3 keV, approximately a factor of 2 higher than achieved in previous experiments with open geometry targets, approaches plasma conditions of high-fusion yield hohlraums. In this new temperature regime, we measure 2omega laser-beam transmission approaching 80% with simultaneous backscattering losses of less than 10%. These findings suggest that good laser coupling into fusion hohlraums using 2omega light is possible.