The discovery and classification of periodic orbits is fundamental to understanding chaotic dynamical systems, but existing algorithms typically search for individual orbits without considering underlying structure and connectivity. We consider the loss landscape of variational loops in phase space, devising a Hessian-based approach to numerically continue along periodic orbit families. Our method offers precise initializations of oscillations around unstable fixed points, an integrator-free variational continuation method, and efficient detection of orbit family intersections and subharmonic bifurcations. Leveraging autograd for computations, we present full continuations of periodic double pendulum oscillations from fixed points, demonstrate examples of orbit family intersections and bifurcations, and interpret branching orbits as combinations of perturbations in the periodic orbit structure.