We present the detection of 1617 new transiting-planet candidates, identified in the Transiting Exoplanet Survey Satellite (TESS) full-frame images observed during the Primary Mission (Sectors 1-26). These candidates were initially detected by the Quick-Look Pipeline (QLP), which extracts full-frame image lightcurves for, and searches all stars brighter than, TESS magnitude T=13.5mag in each sector. However, QLP heavily relies on manual inspection for the identification of planet candidates, limiting vetting efforts to planet-hosting stars brighter than T=10.5mag and leaving millions of potential transit signals unvetted. We describe an independent vetting pipeline applied to QLP transit search results, incorporating both automated vetting tests and manual inspection to identify promising planet candidates around these fainter stars. The new candidates discovered by this ongoing project will allow TESS to significantly improve the statistical power of demographic studies of giant, close-in exoplanets.