Polymer radii of gyration in isotropic single-walled carbon nanotube (SWCNT)/polymer nanocomposites were previously found to increase with increasing SWCNT concentration. Here, the SWCNTs in nanocomposites were aligned by melt fiber spinning, and the polymer chain conformations were found to be anisotropic. Using SAXS and SANS, the anisotropic SWCNT meshes were found to have smaller mesh sizes in the direction perpendicular to the alignment direction than along the alignment direction. At fixed SWCNT orientation, the radius of gyration was probed parallel and perpendicular to the alignment direction, Rgpar and Rgper, respectively, using SANS. With increasing SWCNT concentration, Rgper increases significantly more than Rgpar, such that the extent of anisotropy increases with SWCNT concentration. The anisotropic polymer conformation is larger in the direction perpendicular to the alignment direction, which corresponds to a smaller SWCNT mesh size. Thus, when the SWCNT concentration and alignment combine to produce a SWCNT mesh size that is smaller than the unperturbed Rg, the polymer conformation circumvents the SWCNTs by adopting a larger Rg. Changes in the polymer conformation in nanocomposites with rod-like nanoparticles has important ramifications for entanglement density, polymer dynamics, and mechanical properties.