There are a few ways to track down a release date, but the situation you are describing raises a question worth sitting with first.
Most state and federal prison systems have an inmate locator on their department of corrections website that displays basic information including projected release dates. That is your fastest starting point. Search the DOC website for whatever state he is in, plug in his name or inmate number, and see what comes up. Many of those systems display release dates publicly.
If the website does not show it, call the facility and ask to speak with his counselor or case manager. They may share the release date with an approved contact, though not all facilities will give that information to someone who is not on an official notification list.
Now for the harder part. The fact that he is approved for release, knows the date, and will not tell you on the phone is worth paying attention to. That is not a communication problem; it is a choice. Inmates who want to be picked up by someone tell them when to show up. Someone who is vague about their release date when they have the information may have other arrangements in place, whether that is another person picking them up, a plan they have not shared with you, or simply a situation they are not ready to explain.
Get the date through official channels so you have the information regardless. What you do with it from there is up to you.