  A high-statistics determination of the differential cross section of elastic
muon-electron scattering as a function of the transferred four-momentum
squared, $d \sigma_{el}(\mu e \to \mu e)/dq^2$, has been argued to provide an
effective constraint to the hadronic contribution to the running of the
fine-structure constant, $\Delta \alpha_{had}$, a crucial input for precise
theoretical predictions of the anomalous magnetic moment of the muon. An
experiment called ``MUonE'' is being planned at the north area of CERN for that
purpose. We consider the geometry of the detector proposed by the MUonE
collaboration and offer a few suggestions on the layout of the passive target
material and on the placement of silicon strip sensors, based on a fast
simulation of elastic muon-electron scattering events and the investigation of
a number of possible solutions for the detector geometry.
