We investigate the contribution of star-forming galaxies to the ionizing background at z~3, building on previous work based on narrowband (NB3640) imaging in the SSA22a field. We use new Keck/LRIS spectra of Lyman break galaxies (LBGs) and narrowband-selected Ly{alpha} emitters (LAEs) to measure redshifts for 16 LBGs and 87 LAEs at z>3.055, such that our NB3640 imaging probes the Lyman-continuum (LyC) region. When we include the existing set of spectroscopically confirmed LBGs, our total sample with z>3.055 consists of 41 LBGs and 91 LAEs, of which 9 LBGs and 20 LAEs are detected in our NB3640 image. With our combined imaging and spectroscopic data sets, we critically investigate the origin of NB3640 emission for detected LBGs and LAEs. We remove from our samples three LBGs and three LAEs with spectroscopic evidence of contamination of their NB3640 flux by foreground galaxies and statistically model the effects of additional, unidentified foreground contaminants. The resulting contamination and LyC-detection rates, respectively, are 62%+/-13% and 8%+/-3% for our LBG sample, and 47%+/-10% and 12%+/-2% for our LAE sample. The corresponding ratios of non-ionizing UV to LyC flux density, corrected for intergalactic medium (IGM) attenuation, are 18.0^+34.8^_-7.4_ for LBGs and 3.7^+2.5^_-1.1_ for LAEs. We use these ratios to estimate the total contribution of star-forming galaxies to the ionizing background and the hydrogen photoionization rate in the IGM, finding values larger than, but consistent with, those measured in the Ly{alpha} forest. Finally, the measured UV to LyC flux-density ratios imply model-dependent LyC escape fractions of f^LyC^_esc_~5%-7% for our LBG sample and f^LyC^_esc_~10%-30% for our fainter LAE sample.