Like hydrogen, intergalactic singly ionized helium can be probed by Lyman alpha absorption spectra up to its inferred reionization epoch at z~3. So far, our understanding of HeII reionization has been limited by the handful of z>2 quasars whose far UV flux is not extinguished by optically thick hydrogen Lyman limit systems. However, the search for UV-transparent quasars has become much more efficient with the recent completion of the first UV all-sky survey by the GALEX satellite. We correlated all known z>2.7 quasars with GALEX UV source lists to select new promising sightlines to detect intergalactic HeII. We find that most known quasars with (without) flux down to the onset of intergalactic HeII absorption have blue (red) GALEX UV colors, as expected for sightlines without (with) optically thick intervening absorbers. Artificial GALEX photometry on simulated spectra yields the probability to select transparent sightlines as a function of GALEX UV color and redshift. Eight highly promising UV-bright quasars at 2.73<z<3.15 are scheduled for follow-up spectroscopy with HST/COS in Cycle 17. This first comprehensive sample of new HeII spectra at COS resolution probing similar redshifts will enable a systematic investigation of the HeII reionization epoch. Approved optical echelle spectroscopy will provide the corresponding HI absorption spectra. By comparing HI and HeII absorption we will be able to constrain the spectral shape of the UV background and its fluctuations near the epoch of HeII reionization. With the right targets selected by GALEX, the high UV sensitivity of HST/COS, and the co-spatial information on HI and HeII, it will soon be possible to constrain the HeII reionization epoch at z~3 significantly better than the HI reionization epoch at z>6.