Collective motion of chemotactic bacteria as E. Coli relies, at the individual level, on a continuous reorientation by runs and tumbles. It has been established that the length of run is decided by a stiff response to temporal sensing of chemical cues along the pathway. We analyze consequences on self-organized pattern formation resulting from modulation of tumbling frequency with stiff response. Thanks to both analytical arguments and numerical simulations, we show that the stationary homogeneous state of population density is destabilized in realistic ranges of parameters. A remarkable property is that the unstable frequencies remain bounded, as it is the case in Turing instability. These theoretical developments are in accordance with several biological observations.