  Robust gauge conditions are critically important to the stability and
accuracy of numerical relativity (NR) simulations involving compact objects.
Most of the NR community use the highly robust---though
decade-old---moving-puncture (MP) gauge conditions for such simulations. It has
been argued that in binary black hole (BBH) evolutions adopting this gauge,
noise generated near adaptive-mesh-refinement (AMR) boundaries does not
converge away cleanly with increasing resolution, severely limiting
gravitational waveform accuracy at computationally feasible resolutions. We
link this noise to a sharp (short-wavelength), initial outgoing gauge wave
crossing into progressively lower resolution AMR grids, and present
improvements to the standard MP gauge conditions that focus on stretching,
smoothing, and more rapidly settling this outgoing wave. Our best gauge choice
greatly reduces gravitational waveform noise during inspiral, yielding less
fluctuation in convergence order and $\sim 40%$ lower waveform phase and
amplitude errors at typical resolutions. Noise in other physical quantities of
interest is also reduced, and constraint violations drop by more than an order
of magnitude. We expect these improvements will carry over to simulations of
all types of compact binary systems, as well as other $N$+1 formulations of
gravity for which MP-like gauge conditions can be chosen.
