Until very recently, people throughout the league believed that the procedure for returning from the reserve/left squad list was administrative, and essentially automatic.
The Brandon Aiyuk case shows that, in the NFL's view, it definitely is not.
The Collective Bargaining Agreement is silent on the topic of the reserve/left squad list. The relevant language regarding placement (and eventual reinstatement) from the reserve/left squad list appears in the NFL's Player Personnel Handbook.
In the year when the player is placed on reserve/left squad, he can't play for his current team or any other team. His season is basically over. The window for reinstatement opens the next year.
Here's the key language from the Player Personnel Handbook regarding reinstatement: "A player on Reserve/Left Squad must apply to the Commissioner for reinstatement to his club's Active List. No player on Reserve/Left Squad shall be reinstated in the same season in which he is placed on Reserve. A player applying for reinstatement in a season subsequent to the season in which he is placed on Reserve/Left Squad is prohibited from being reinstated...