The 3rd flight of the Focusing Optics X-ray Solar Imager (FOXSI-3) sounding rocket will perform a high-sensitivity search for small-scale energy release in the quiet and active Sun. Small energy releases in the corona (nanoflares) have the possibility to heat the solar corona to its high observed temperature. Nanoflares can be investigated via direct hard X-ray (HXR) observation of flare-accelerated electrons or by observation of hot, flare-temperature plasma in the otherwise quiescent corona. Unlike previous solar HXR instruments that relied on indirect Fourier imaging, FOXSI uses direct focusing HXR optics. FOXSI has successfully flown twice (Nov. 2, 2012 and Dec. 11, 2014), with a major aim of investigating quiet-Sun HXR emission. However, all FOXSI flights, as well as all NuSTAR solar observations, have been background-limited for quiet Sun nanoflare investigation by single-bounce photons from sources outside the field of view. A major improvement to the FOXSI-3 payload will be a collimating structure that blocks single-bounce flux. Other upgrades will include the replacement of several detectors with improved CdTe detectors provided by ISAS/JAXA and a fuller set of mirrors in each module. With these upgrades, the FOXSI-3 payload will (1) perform a high-sensitivity search for nonthermal electrons in the quiet Sun, (2) measure active region DEMs with better energy coverage than that of NuSTAR (due to deadtime restrictions), and (3) develop a collimation system essential for future direct solar HXR imagers.