It has been recently shown that for quantum systems with dimension d >= 4 unknown pure states can be estimated via measurements on five bases only. Here, we study by means of numerical experiments the estimation accuracy of this method as a function of the dimension and the number of independently, identically prepared copies, that is, ensemble size. We show that the accuracy of this method can be greatly improved by modifying the estimation procedure, without increasing the number of measurement outcomes. The present estimation accuracy becomes approximately d times smaller than the best accuracy achievable by tomographic methods for unknown mixed states. We also study the case of pure states affected by a low level of white noise. We show that the modified version of the method is very robust; that is, it remains to a large extent unaffected by the noise and achieves an accuracy similar to the case without noise.