Coronal Mass Ejections (CMEs), commonly referred to as solar storms, that are on an Earth-intersecting trajectory, may lead to the breakdown of power grid transformers, the malfunctioning of Earth-orbiting spacecraft, and disruptions in navigation and communication systems, among many other effects. The financial impact of a solar storm is predicted to be in the order of trillions of euros and the probability of such an event occurring within the next decade is 12%. With society relying ever-more on technology, the impact of a solar storm is ever-increasing. It is therefore essential that operators of vital infrastructure are notified of an approaching storm in a timely manner such that they can take adequate measures to mitigate the impact. This paper investigates the use of solar-sail technology to increase the warning time for CMEs heading towards Earth. The warning time is proportional to the distance from the Earth to the spacecraft detecting the CME: a current warning time of 30 to 60 minutes is achieved by satellites at or near the Sun-Earth L1 point. By considering the actual shape of a CME, the continuous solar-sail acceleration from the solar sail can be used to find a periodic trajectory that travels further upstream of the CME-axis, thereby increasing the warning time with respect to current missions. Finding a periodic solar-sail trajectory can be regarded as an optimal control problem, which requires a near-feasible initial-guess trajectory. The latter is found by generating heteroclinic connections between artificial equilibrium points in the vicinity of the sub-L1 and sub-L5 point through the use of a genetic algorithm. The optimal control problem is solved with a direct pseudospectral method, resulting in four representative trajectories, each having specific (dis)advantages. Ultimately, with near-term solar-sail technology (a lightness number of 0.05), the most optimal trajectory increases the average and maximum warning time by a factor 20 and 30 with respect to current missions at L1, respectively, with a 90% probability that the spacecraft detects the CME. Finally, the paper investigated a set of sensitivity analyses (non-ideal sail properties and change in lightness number) to successfully prove the robustness of the methodology and the effect of assumptions made.