The particle beam self-modulation instability in tapered and inhomogeneous plasmas is analyzed via an evolution equation for the beam radius. For a sufficiently fast taper, the instability is suppressed, and the condition for growth suppression is derived. The form of the taper to phase lock a trailing witness bunch in the plasma wave driven by a self-modulated beam is determined, which can increase the energy gain by several orders of magnitude. Growth of the instability places stringent constraints on the initial background plasma density fluctuations.