Multi-spacecraft Solar Energetic Particle (SEP) observations show that the SEPs accelerated during solar eruptions have fast access to large distances across the mean Parker spiral field. SEP anisotropy measurements suggest interplanetary transport strongly contributes to the cross-field extent of SEP events. However, the currently used diffusive Fokker Planck (FP) models of SEP transport cannot explain the observed SEP events with realistic cross-field diffusion coefficients. Recently Laitinen et al (2013) showed the importance of particle propagation along meandering field lines, which cannot be described as diffusion early in the event history. Their result suggested that, due to field line meandering, the SEP event extent can be expected to be wider than that predicted by the standard FP models. In this work, we implement a new FP model that incorporates field line meandering in a Parker magnetic field geometry, to study the evolution of an SEP event in the heliosphere, and compare the new model to the traditional FP approach. The particle and field line diffusion coefficients are calculated using an assumed radial and spectral description of the turbulence evolution. Our new model results in very wide SEP events, with $\sigma=33^\circ$ for Gaussian helio-longitude distribution of SEP peak intensities, consistent with multi-spacecraft SEP observations by STEREO spacecraft. This far exceeds the standard FP model extents of $\sigma=10^\circ$. We conclude that field line meandering is a necessary factor in determining particle access to widely separated spacecraft locations.