Absorption along quasar sightlines remains among the most sensitive direct measures of HeII reionization in much of the intergalactic medium (IGM). Until recently, fewer than a half-dozen unobscured quasar sightlines suitable for the HeII Gunn-Peterson test were known; although these handful demonstrated great promise, the small sample size limited confidence in cosmological inferences. We have recently added nine more such clean HeII quasars, exploiting Sloan Digital Sky Survey (SDSS) quasar samples, broadband ultraviolet (UV) imaging from Galaxy Evolution Explorer (GALEX), and high-yield UV spectroscopic confirmations from Hubble Space Telescope (HST). Here we markedly expand this approach by cross-correlating SDSS DR7 and GALEX GR4+5 to catalog 428 SDSS and 165 other quasars with z>2.78 having likely (~70%) GALEX detections, suggesting they are bright into the far-UV. Reconnaissance HST Cycle 16 Supplemental prism data for 29 of these new quasar-GALEX matches spectroscopically confirm 17 as indeed far-UV bright. At least 10 of these confirmations have clean sightlines all the way down to HeII Ly{alpha}, substantially expanding the number of known clean HeII quasars, and reaffirming the order of magnitude enhanced efficiency of our selection technique. Combined confirmations from this and our past programs yield more than 20 HeII quasars, quintupling the sample. These provide substantial progress toward a sample of HeII quasar sightlines large enough, and spanning a sufficient redshift range, to enable statistical IGM studies that may avoid individual object peculiarity and sightline variance. Our expanded catalog of hundreds of high-likelihood far-UV-bright QSOs additionally will be useful for understanding the extreme-UV properties of the quasars themselves.