We study the nonlinear dynamics of hydrothermal waves produced by a surface-tension-driven convective flow in a long and thin annular channel heated from the side. Above onset, the supercritical traveling wave pattern undergoes a secondary instability: a supercritical Eckhaus instability. This leads to a small-wave-number phase-modulated nonlinear mode, and shows the first experimental evidence of a nonlinearly saturated phase instability mode for traveling wave patterns. At higher forcing level, this secondary pattern is subject to a tertiary instability. This mode is an amplitude mode characterized by traveling hole patterns, i.e., space-time defects that change the wave number.