A series of waveguides were inscribed in lithium niobate by tightly focused femtosecond-laser pulses of 11-MHz repetition rate and 790-nm wavelength. To establish the inscription conditions for optimal low-loss waveguides, within each sample we varied laser pulse energy, speed and direction of translation stage movement, and focus depth of the beam. We deployed two new approaches to enhance the inscription results: 1) increase of the pulse energy with increasing focus depth inside the material to compensate for the corresponding decrease of refractive-index modification, and 2) decrease of the laser energy for the modification tracks closer to the waveguide’s core region to reduce scattering losses due to high-laser-energy driven non-uniformities. All waveguides had an optical-lattice-like hexagonal packing geometry with track-spacing of 9.9 μm (optimized for effective suppression of high-order modes). Each structure comprised 84 single-scan Type-II-modification tracks, aligned with the crystalline X-axis of lithium niobate. After heat treatment at 350 °C for 3 hours, the lowest propagation loss of less than (0.4±0.1) dB/cm and (3.5±0.3) dB/cm for the ordinary and extraordinary light polarization states, respectively, were achieved at the 1550- nm wavelength. These low-attenuation waveguides were obtained with the inscription energy varying between 50.6 nJ and 53.6 nJ and the translation speed of 10 mm/s. The corresponding refractive-index contrast of individual tracks was (–1.55±0.04)×10<sup>-3</sup> . The waveguides also showed low attenuation in the visible and near-infrared portion of the spectrum (532 nm to 1456 nm). Our results offer promising means for the development of low-loss waveguides with preserved-nonlinearity and high thermal stability.