A detailed assessment is made here of a recently proposed buckling model for a drill string that remains in conformal contact with an axially-toroidal borehole. Suitable rescaling of the relevant equations allows us to identify a particular regime of interest, which is subsequently explored in depth with a multiple-scale asymptotic strategy. In particular, the localisation phenomena previously reported in the literature are placed on firm ground and explained with the help of a small number of analytical formulae. Two new sets of qualitatively different solutions are also identified and discussed in detail. Our asymptotic predictions of the critical loads are shown to be in very good agreement with the direct numerical simulations of the corresponding linear bifurcation problem.