This paper investigates the use of solar-sail technology to increase the warning time for Coronal Mass Ejections (CMEs) heading towards Earth. In addition, this research will build upon the current understanding of using solar-sail dynamics with regards to CME detection by providing insights into the problem characteristics. 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 grid search and a genetic algorithm. The optimal control problem is solved with a direct pseudospectral method, resulting in four representative trajectories, each having specific (dis)advantages. The performance impact due to (the uncertainty of) non-ideal sail properties, change in lightness number, and variation in CME size are investigated. Ultimately, 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.