The Hiring Manager is set to either a Job Owner or a Hiring Manger at the job level here:
Based on how they are designated there, they might not have access to all of the stages. You can check the job to see how they are set and then check the workflow stages via Setup--> Configure --> Configure Application Workflow, then edit the workflow that is attached tot he job, to see it they as a group (Job Owners and/or Hiring Managers) have access to the stage.
The following example shows that only the Job Owners group (set at the job level) have access to the New stage, but the Manager Review stage has access to both the Job Owner & Hiring Manger Groups:
You can either edit the job to give the user access to the job as a Job Owner or you can edit the stage to give Hiring Mangers access to the stage.