We consider a system in which a planar shock propagates into a plasma where the magnetic field consists of a uniform background component that is nearly parallel to the shock surface (i.e., the shock is nearly perpendicular) plus a random component that is transverse to the background field. The random component is a superposition of Alfvén waves with a wide range of wavelengths and with amplitudes determined from a power spectral density function that is flat at long wavelengths and a power law at short wavelengths. We investigate the characteristics of superthermal ion (proton) distributions accelerated in this system by integrating along particle orbits. The main difference between results reported herein and those reported previously by us is the major role played by magnetic loops in mediating multiple shock interactions. The transverse magnetic fluctuations are composed of short wavelength components that scatter the ions in pitch angle, and long wavelength components that produce a large‐scale spatial meandering of magnetic field lines. At nearly perpendicular shocks, this meandering results in a distribution of transient magnetic loops of various sizes at the shock. Charged particles are guided by the large‐scale spatial variations and can interact with the shock at many points, becoming temporarily trapped along several such loops, and also being scattered by resonant waves during and between shock interactions. A few particles can acquire large energy gains before the field lines along which they propagate convect through the shock. As a specific application of the model, we investigate the acceleration of a monoenergetic seed proton population for both impulsive and continuous time injection at a shock that is on average perpendicular. In the model the typical energy density of the magnetic fluctuations is 10–20% that of the background field. Characteristic features of the accelerated proton distributions are (1) energy spectra with power law tails extending to at least 1000 times the injection energy, with spectral slopes somewhat steeper than those predicted, for example, by diffusive shock acceleration theory; (2) upstream intensity precursors that grow exponentially towards the shock, withe‐folding scales that increase with energy up to a maximum scale of the order of the spatial scale of the meandering field lines; (3) upstream angular distributions with large field‐aligned bidirectional anisotropies throughout the intensity precursor; (4) downstream angular distributions with weaker anisotropies peaked transverse to the magnetic field within an intensity plateau extending from the shock to the convection boundary, and field‐aligned bidirectional distributions beyond the convection boundary. We develop a simplified model that explains the salient features of the upstream precursors in terms of the statistical properties of the long‐wavelength components that comprise the random field and suggest how electron acceleration can be accommodate