Quantum state teleportation is an important protocol that plays a pivotal role in various quantum information tasks. Here we theoretically investigate quantum state teleportation by exploiting a general two-mode Gaussian entangled state produced by a parametric converter when two single-mode Gaussian states (SMGSs) in terms of the non-classicality and purity are employed as inputs to the parametric converter. In particular, the time-dependent teleportation fidelity is analyzed with respect to the squeezing parameter and phase-space quadratures of the teleported squeezed coherent state. We show that the teleportation fidelity is maximal when ratio of the expectation values of photon number in the two modes of the evolved Gaussian entangled state is equal to 1. Quantum state teleportation in terms of the purity and non-classicality of the general two SMGSs seems to be a good choice for experimental realization of quantum communication and information processing.