Before revision:  Our  numerical results show that first-order term typically exhibits rather poor performance and that the second-order term significantly improves the approximation's accuracy.
After revision:  Our method of proof is based on an integral representation of the option price involving the tail probability of the log-return process under the share measure and a suitable change of probability measure under which the pure-jump component of the log-return process becomes a Y-stable process. Our approach is sufficiently general to cover a wide class of L\'evy processes which satisfy the latter property and whose L\'evy densitiy can be closely approximated by a stable density near the origin. Our numerical results show that first-order term typically exhibits rather poor performance and that the second-order term significantly improves the approximation's accuracy.