This paper aims to propose convex-optimization-based entry guidance for a spaceplane, which has potential in online implementation with less sensitivity to initial guess accuracy while mitigating a high-frequency jittering issue in the entry trajectory optimization problem. To this end, a highly nonlinear, constrained, and nonconvex entry guidance problem is converted into sequential convex sub-problems in the second-order cone programming (SOCP) form by an appropriate combination of successive linearization and convexification techniques. From the investigation on the potential sub-problem infeasibility due to a rough initial guess for radial distance, a linear penalized term associated with a virtual control for an inequality constraint is used to relieve the sub-problem infeasibility while preserving the standardized SOCP form. An adjustable trust-region bound is also adopted in the proposed approach to improve the convergence property further. Additionally, a change of control variables and a relaxation technique are utilized to relieve the high-frequency jittering issue. It is proven that the Lossless convexification property is preserved for the relaxed problem even in the presence of the penalty terms. The feasibility of the proposed method is investigated through numerical simulations.