Grace, Jet, and Oswald went into the hut to rest. Jack didn't go to sleep immediately. Despite it being almost midnight, he still wanted to do his regular training before bed. It didn't sit right with him to skip a training session, not to mention the mana density here greatly improved his learning process. That's why he had asked Whitebeard if they were allowed outside the hut. He couldn't train inside without disturbing the others.
He cast his spells and trained his mana manipulation on the spells. He also cast spells to increase their proficiencies. Skills as well, especially the Spirit Weapon skill. Peniel mentioned that with each star, the spirit weapon would get more powerful.