The main plot of Judgment is rather extensive and divided into chapters. In addition to the main tasks, you can find side activities such as mini-games or additional investigations. Read this chapter to find out whether you can return to these activities after you beat the main storyline.
Quick answer - yes, you can play and enjoy the additional content of the game. Judgment is designed in such a way that would let players enjoy the title as long as possible. Besides fights and the main storyline, the game also has a lot of side activities and challenges. After completing the story, you can explore Kamurocho and participate in mini-games to complete cases or to establish relationships with friends.
An added bonus is that you can sleep in Yagami's apartment to change the time of day, e.g. go from day to night. This is useful because some tasks require a certain time of the day. So it is up to you how you want to experience the content of Judgment - you can complete cases and side activities as your progress through the main storyline, or wait for the story to end and then complete these tasks.