In this paper we study an asymptotic version of the holonomic efficiency problem which originated in the study of swimming microorganisms. Given a horizontal distribution on a vector bundle, the holonomy of a loop in the base space is the displacement along the fiber direction of the end points of its horizontal lift. The holonomic efficiency problem is to find the most efficient loop in the base space in terms of gaining holonomy, where the cost of the base loop is measured by a subriemannian metric, and the holonomy gained is compared using a test function. We introduce the notions of rank and asymptotic holonomy and characterize them through the series expansions of holonomy as a function of the loop scale. In the rank two case we prove that for convex test functions the most efficient base loops are simple circles, and we solve these loops for linear and norm test functions. In the higher rank case the analytical solutions are outlined for some special instances of the problem. An example of a turning linked-mass system is worked out in detail to illustrate the results.