The conditions derived in [K. Scherer and P. W. Smith, SIAM J. Numer. Anal., 20 (1989), pp. 160--168] for the existence of minimizers to the nonlinear problem of best "interpolation" by curves are extended to the problem of best "near-interpolation" by curves that meet arbitrary sets, such as closed balls (as in [S. Kersey, Best Near-Interpolation by Curves: Optimality Conditions, Technical Report 99-05, Center for the Mathematical Sciences, University of Wisconsin, Madison, WI, 1999]). The minimizers are spline curves with breakpoints at the data sites at which the curves meet the sets, and the nonlinearities arise as these data sites vary from curve to curve. The results here apply to Hermite-type interpolation conditions, with the possibility of repeated data sites.