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We consider a system of stacked tunnel-coupled two-dimensional electron- and hole-gas layers with Rashba spin-orbit interaction subjected to a staggered Zeeman field. The interplay of different intralayer tunnel couplings results in a topological insulator phase characterized by the presence of gapless helical surface states. The staggered Zeeman field further enriches the topological phase diagram by generating a second-order topological insulator phase characterized by the presence of gapless hinge states. The emergence of these topological phases is proven analytically in the regime of small Zeeman field and confirmed by numerical simulations in the nonperturbative region of the phase diagram. The topological phases are stable against external perturbations and disorder.