We prove that certain non-exact magnetic Hamiltonian systems on products of closed hyperbolic surfaces and with a potential function of large oscillation admit non-constant contractible periodic solutions of energy below the Mañé critical value. For that we develop a theory of holomorphic curves in symplectizations of non-compact contact manifolds that arise as the covering space of a virtually contact structure whose contact form is bounded with all derivatives up to order three.

Motivated by recent developments in proving the Weinstein conjecture we introduce the notion of covering contact connected sum for virtually contact manifolds and construct virtually contact structures on boundaries of subcritical handle bodies.

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