We consider a local version of the assignment problem for the dichotomy spectrum of linear continuous time-varying systems defined on the half-line. Our aim is to show that uniform complete controllability is a sufficient condition to place the dichotomy spectrum of the closed-loop system in an arbitrary position within some Hausdorff neighborhood of the dichotomy spectrum of the free system using an appropriate time-varying linear feedback. Moreover, we assume that the norm of the matrix of the linear feedback should be bounded from above by the Hausdorff distance between these two spectra with some constant multiplier.