We consider DPG methods with optimal test functions and broken test spaces based on ultra-weak formulations of general second-order elliptic problems. Under some assumptions on the regularity of solutions of the model problem and its adjoint, superconvergence for the scalar field variable is achieved by either increasing the polynomial degree in the corresponding approximation space by one or by a local postprocessing. We provide a uniform analysis that allows the treatment of different test norms. Particularly, we show that in the presence of convection only the quasi-optimal test norm leads to higher convergence rates, whereas other norms considered do not. Moreover, we also prove that our DPG method delivers the best L-2 approximation of the scalar field variable up to higher-order terms, which is the first theoretical explanation of an observation made previously by different authors. Numerical studies that support our theoretical findings are presented.