In many-body chaotic systems, the size of an operator generically grows in Heisenberg evolution, and is measured by out-of-time-ordered four-point functions. However, these only provide a coarse probe of the full underlying operator growth structure. We develop a methodology to derive the full growth structure of fermionic systems, that also naturally introduces the effect of finite temperature. We then apply our methodology to the SYK model, which features all-to-all q-body interactions. We derive the full operator growth structure in the large q limit at all temperatures. We see that its temperature dependence has a remarkably simple form consistent with the slowing down of scrambling as temperature is decreased. Furthermore, the finite-temperature scrambling follows a modified epidemic model, where the thermal state serves as a vaccinated population, thereby slowing the overall rate of infection. Finally and perhaps most interestingly, we note that at strong coupling/low temperatures the cascade into larger operators mirrors an AdS2-Rindler probe particle's growth into states of larger global energy or null momentum.