  We construct accurate models of three-nucleon (3N) interaction by fitting, in
a hybrid phenomenological approach, the low-energy constants parametrizing the
subleading 3N contact operators to the triton binding energy, n-d scattering
lengths, cross section and polarization observables of p-d scattering at 2 MeV
center-of-mass energy. These models lead to a satisfactory description of
polarized p-d scattering data in the whole energy range below the deuteron
breakup threshold. In particular, the long-standing $A_y$ puzzle seems to be
solved thanks to the new terms considered in the 3N force. Two types of
hierarchies among the subleading contact operators are also derived, based on
the large-$N_c$ counting and on a recently proposed relativistic counting. We
test these hierarchies against the same experimental data and show that they
are respected at a reasonable level.
