  Numerical simulations of binary black holes are accompanied by an initial
spurious burst of gravitational radiation (called `junk radiation') caused by a
failure of the initial data to describe a snapshot of an inspiral that started
at an infinite time in the past. A previous study showed that the superposed
harmonic (SH) initial data gives rise to significantly smaller junk radiation.
However, it is difficult to construct SH initial data for black holes with
dimensionless spin $\chi\gtrsim0.7$. We here provide a class of spatial
coordinate transformations that extend SH to higher spin. The new spatial
coordinate system, which we refer to as superposed modified harmonic (SMH), is
characterized by a continuous parameter -- Kerr-Schild and harmonic spatial
coordinates are only two special cases of this new gauge. We compare SMH with
the superposed Kerr-Schild (SKS) initial data by evolving several binary black
hole systems with $\chi=0.8$ and $0.9$. We find that the new initial data still
leads to less junk radiation and only small changes of black hole parameters
(e.g. mass and spin). We also find that the volume-weighted constraint
violations for the new initial data converge with resolution during the junk
stage $(t\lesssim700M)$, which means there are fewer high-frequency components
in waveforms at outer regions.
