The effects of horizontal pressure gradient and Soret coefficient on the onset of double-diffusive convection in a fluid-saturated porous layer under the influence of local thermal nonequilibrium (LTNE) temperatures are analyzed. Darcy's law with local acceleration term, which involves the two-field temperature model describing the fluid and solid phases separately and the approximation of Oberbeck-Boussinesq, is used. The dynamics of small-amplitude perturbations on the basic mixed convection flow is studied numerically. Using the Galerkin method along with the QZ-algorithm, the eighth order eigenvalue differential equation obtained by employing linear stability analysis is solved. The solution provides the neutral stability curves and determines the threshold of linear instability, and the critical values of thermal Darcy-Rayleigh number, wave number, and the frequency at the onset of instability are determined for various values of control parameters. It is found that, rather than the stationary motion, the instability is found to be via oscillatory motion. Besides, the contribution to each parameter on stability characteristics is explored in detail, and some relevant findings have been described that have not been reported hitherto in the literature.