We recently extended our Parker-type transport equation for energetic particle interaction with numerous dynamic small-scale magnetic flux ropes (SMFRs) to include perpendicular diffusion in addition to parallel diffusion. We present a new analytical solution to this equation assuming heliocentric spherical geometry with spherical symmetry for all SMFR acceleration mechanisms present in the transport theory. With the goal of identifying the dominant mechanism(s) through which particles are accelerated by SMFRs, a search was launched to identify events behind interplanetary shocks that could be explained by our new solution and not classical diffusive shock acceleration. Two new SMFR acceleration events were identified in situ for the first time within heliocentric distances of 1 astronomical unit (au) in Helios A data. A Metropolis–Hastings algorithm is employed to fit the new solution to the energetic proton fluxes so that the relative strength of the transport coefficients associated with each SMFR acceleration mechanism can be determined. We conclude that the second-order Fermi mechanism for particle acceleration by SMFRs is more important than first-order Fermi acceleration due to the mean compression of the SMFRs regions during these new events. Furthermore, with the aid of SMFR parameters determined via the Grad–Shafranov reconstruction method, we find that second-order Fermi SMFR acceleration is dominated by the turbulent motional electric field parallel to the guide/background field. Finally, successful reproduction of energetic proton flux data during these SMFR acceleration events also required efficient particle escape from the SMFR acceleration regions.