A model for high-T(c) oxides, consisting of Josephson-coupled superconducting layers with both intra- and interlayer pairings, is used to study the anomalous positive curvature in the temperature dependence of the lower critical magnetic field, H(c1)(T). An equilibrium distribution of magnetic fields in a single vortex is calculated for magnetic field parallel and perpendicular to the superconducting layers. By using this distribution the lower critical field H(c1) is obtained. For Josephson-coupled layered superconductors with only intralayer pairing, the cross section of the single vortex parallel to the layers is known to have an elliptic form. The existence of two order parameters in our model gives rise to an additional anisotropy in the distribution of the magnetic field in a single vortex. The results obtained for H(c1) make it possible to understand the experimentally observed anomalous enhancement of H(c1) at low temperatures in YBa2Cu3O7, as well as in Tl- and Bi-based superconductors.