This paper is concerned with improvements in some exact formulae for the period of the simple pendulum problem. Two recently presented formulae are re-examined and refined rationally, yielding more accurate approximate periods. Based on the improved expressions here, a particular new formula is proposed for the period. It is shown that the derived formula has considerably high accuracy for all amplitudes of the pendulum, resulting in a maximum percentage relative error of only 0.035. (Contains 2 figures.)