Push and pull magnetic reconnection (MR) experiments using high-power laser irradiating a capacitor target with the plates connected by a pair of U-shaped coils are carried out. During the beginning (end) of the laser-target interaction that creates a hot plasma in the region, the rise (fall) stages of the coil currents generates expanding (contracting) magnetic fields that reconnect in the midplane between the coils, resulting in push (pull) MR. Proton radiography and proton ray-tracing simulation are used to track the evolution of the magnetic fields. The proton accumulation and void formation between the coils are related to the oppositely directed current-sheet currents during the push and pull MR stages. The directions of the plasma electron outflows during these two MR phases are obtained by monitoring the soft x-ray emission. Our results suggest that the double-coil capacitor target may be useful for laboratory modeling of fast MR and related phenomena in astrophysical plasmas.