Precise collimation of the beam halo is required in the International Linear Collider (ILC) to prevent beam losses near the interaction region that could cause unacceptable backgrounds for the physics detector. The necessarily small apertures of the collimators lead to transverse wakefields that may result in beam deflections and increased emittance. A set of collimator wakefield measurements has previously been performed in the ASSET region of the SLAC Linac. We report on the next phase of this programme, which is carried out at the recently commissioned End Station A (ESA) test facility at SLAC. Measurements of resistive and geometric wakefields using tapered collimators are compared with model predictions from MAFIA and GdfidL and with analytic calculations.