  The electric dipole moment of the electron (eEDM) can be measured with high
precision using heavy polar molecules. In this paper, we report on a series of
new techniques that have improved the statistical sensitivity of the YbF eEDM
experiment. We increase the number of molecules participating in the experiment
by an order of magnitude using a carefully designed optical pumping scheme. We
also increase the detection efficiency of these molecules by another order of
magnitude using an optical cycling scheme. In addition, we show how to
destabilise dark states and reduce backgrounds that otherwise limit the
efficiency of these techniques. Together, these improvements allow us to
demonstrate a statistical sensitivity of $1.8 \times 10^{-28}$ e cm after one
day of measurement, which is 1.2 times the shot-noise limit. The techniques
presented here are applicable to other high-precision measurements using
molecules.
