Let Ω be a subdomain of a Stein surface with smooth strictly pseudoconvex boundary M=∂Ω. We are mainly interested in the case that Ω is not relatively compact. Building on work by G. Lupacciolu, we describe the envelope of holomorphy of an arbitrary open subset M∗ of M in terms of a modified holomorphic hull of the complement A=M\M∗, designed in order to also reflect the geometry of Ω at infinity. As a consequence, we clarify the relation between the envelope of M∗ and some notions of core sets of Ω, which are an active topic in recent research.