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In a muon accelerator complex, a target is bombarded by a multi-MW proton beam to produce pions, which decay into the muons which are thereafter bunched, cooled, and accelerated. The front end of the complex captures those pions, then manipulates their phase space, and that of the muons into which they decay, to maximize the number of muons within the acceptance of the downstream systems. The secondary pion beam produced at the target is captured by a high field target solenoid that tapers down to a constant field throughout the rest of the front end. In this study we enhance the useful muon flux by introducing a new design of the longitudinal profile of the solenoid field at, and downstream of, the target. We find that the useful muon flux exiting the front end is larger when the field at the target is higher, the distance over which the field tapers down is shorter, and the field at the end of the taper is higher. We describe how the solenoid field profile impacts the transverse and longitudinal phase space of the beam and thereby leads to these dependencies.