  Seeded Self-modulation (SSM) has been demonstrated to transform a long proton
bunch into many equidistant micro-bunches (e.g., the AWAKE case), which then
resonantly excite strong wakefields. However, the wakefields in a uniform
plasma suffer from a quick amplitude drop after reaching the peak. This is
caused by a significant decrease of the wake phase velocity during
self-modulation. A large number of protons slip out of focusing and
decelerating regions and get lost, and thus cannot contribute to the wakefield
growth. Previously suggested solutions incorporate a sharp or a linear plasma
longitudinal density increase which can compensate the backward phase shift and
therefore enhance the wakefields. In this paper, we propose a new plasma
density profile, which can further boost the wakefield amplitude by 30%. More
importantly, almost 24% of protons initially located along one plasma period
survive in a micro-bunch after modulation. The underlying physics is discussed.
