Once fully rested, Jax wore his new tunic and continued on with the maze. With the 1-Star ticket acquired, he didn't need to worry about failing to find any bosses by the time it all came to an end.
There were less than 4 days remaining at this point, so finding a ticket with 2-Star or higher would be quite difficult. He needed to hurry.
There was no basis on which he decided where he was going to go, so Jax went wherever his whims took him. He walked around for hours on end, spending just a few moments to rest.
He met 2 bosses along the way in the many hours he walked around the maze, but both of them were 1-Star Tickets, so he ended up skipping them. He didn't have the time to spend fighting them.
However, as the entire day passed, Jax realized that if he didn't stop somewhere safe to sleep, he was going to be too drowsy to do anything at all.