  We propose a new radial coordinate to write the Kerr metric in puncture form.
Unlike the quasi-radial coordinate introduced previously, the horizon radius
remains finite in our radial coordinate in the extreme Kerr limit a/M -> 1.
This significantly improves the accuracy of the evolution of black holes with
spins close to the extreme Kerr limit. We are able to evolve accurately both
stationary and boosted black holes with spins as high as a/M=0.99 using initial
data constructed in these new puncture coordinates. Initial data of compact
binaries with rapidly spinning black holes can be constructed using our
proposed new puncture metric for the background conformal metric. Our
simulations for single black holes suggest that such initial data can be
evolved successfully by the moving puncture technique.
