  Observations of reionization-era analogs at $z\sim3$ are a powerful tool for
constraining reionization. Rest-ultraviolet observations are particularly
useful, in which both direct and indirect tracers of ionizing-photon production
and escape can be observed. We analyse a sample of 124 $z\sim3$ galaxies from
the Keck Lyman Continuum Spectroscopic Survey, with sensitive spectroscopic
measurements of the Lyman continuum region. We present a method of removing
foreground contamination from our sample using high-resolution, multi-band
Hubble Space Telescope imaging. We re-measure the global properties of the
cleaned sample of 13 individually-detected Lyman continuum sources and 107
individually-undetected sources, including a sample-averaged absolute escape
fraction of $f_{\rm esc,abs}=0.06\pm0.01$ and a sample-averaged ratio of
ionizing to non-ionizing ultraviolet flux density of $<f_{900}/f_{1500}>_{\rm
out}=0.040\pm0.006$, corrected for attenuation from the intergalactic and
circumgalactic media. Based on composite spectra, we also recover a strong
positive correlation between $<f_{900}/f_{1500}>_{\rm out}$ and Ly$\alpha$
equivalent width (W$_\lambda$(Ly$\alpha$)) and a negative correlation between
$<f_{900}/f_{1500}>_{\rm out}$ and UV luminosity. As in previous work, we
interpret the relationship between $<f_{900}/f_{1500}>_{\rm out}$ and
W$_\lambda$(Ly$\alpha$) in terms of the modulation of the escape of ionizing
radiation from star-forming galaxies based on the covering fraction of neutral
gas. We also use a W$_\lambda$(Ly$\alpha$)-weighted $<f_{900}/f_{1500}>_{\rm
out}$ to estimate an ionizing emissivity from star-forming galaxies at $z\sim3$
as $\epsilon_{\rm LyC}\simeq5.5\times10^{24}$erg s$^{-1}$ Hz$^{-1}$ Mpc$^{-3}$.
This estimate, evaluated using the uncontaminated sample of this work,
reaffirms that galaxies provide the majority of the ionizing background at
$z\sim3$ and beyond.
