The optically and IR-bright and starlight-scattering HR 4796A ringlike debris disk is one of the most-(and best-) studied exoplanetary debris systems. The presence of a yet-undetected planet has been inferred (or suggested) from the narrow width and inner/outer truncation radii of its r = 1.'' 05 (77 au) debris ring. We present new, highly sensitive Hubble Space Telescope (HST) visible-light images of the HR 4796A circumstellar debris system and its environment over a very wide range of stellocentric angles from 0.'' 32 (23 au) to approximate to 15 '' (1100 au). These very high-contrast images were obtained with the Space Telescope Imaging Spectrograph (STIS) using six-roll PSF template-subtracted coronagraphy suppressing the primary light of HR 4796A, with three image-plane occulters, and simultaneously subtracting the background light from its close angular proximity M2.5V companion. The resulting images unambiguously reveal the debris ring embedded within a much larger, morphologically complex, and biaxially asymmetric exo-ring scattering structure. These images at visible wavelengths are sensitive to and map the spatial distribution, brightness, and radial surface density of micron-size particles over 5 dex in surface brightness. These particles in the exo-ring environment may be unbound from the system and interacting with the local ISM. Herein, we present a new morphological and photometric view of the larger-than-prior-seen HR 4796A exoplanetary debris system with sensitivity to small particles at stellocentric distances an order of magnitude greater than has previously been observed.