One of my player began a hardcover then played a DDAL module before finishing the chapter of the Hardcover. May he gain a level after the module and continue the hardcover with this new level ?
This is answered in the FAQ_v9.1 of the Adventure League Player & DM Pack
Characters can play other adventures between sessions
They can level up and earn rewards playing other adventures between sessions.
Playing Content Between Sessions
Characters can play other adventures (including other multiple-session adventures) between sessions.
Players taking their characters from game to game in this way are permitted to advance in level and earn rewards between sessions of a multiple-session adventure but must exercise caution or else they may accidentally level-out of their original game.
Characters must be careful not to level out of the hardcover level range
Hardcover Adventures A character can only play one hardcover adventure at a time. If character playing one adventure jumps into another hardcover from a different storyline, and levels outside of the first adventure’s level range, they can’t return to the first adventure.