Brian refers to that shift in his life and attitude as "new time" because the shift is a complete turn in Brian's attitude about his situation. People talk about the idea of "turning over a new leaf." It's usually about some habit change that they want to make; however, Brian literally has a completely new outlook on his life.
Before the "new time," Brian's main hope for survival was rescue. As the days ticked by, Brian got hungrier and weaker. Every day he wasn't rescued was another day of lost hope. The new time isn't a time of losing hope. It is a time of Brian having increased hope in his survival. The difference is that Brian knows that his survival is not dependent on somebody else rescuing him. His survival depends on his own ability to avoid costly mistakes and secure himself food.
But hope in his knowledge. Hope in the fact that he could learn and survive and take care of himself. Tough hope, he thought that night. I am full of tough hope.
The "new time" is about Brian's new attitude toward seizing control of his own survival.