How do I get an automated message telling people who call my old phone number my new number?

I’m thinking about switching numbers because I moved out of state and it’s confusing having my old area code.

At my work, we do confirmation calls, and once, I called someone and I got an automated message saying "The number you have dialed has changed. The new number is: … This number will be repeated once. The new number is: … Thank you." And then it hung up. I thought it was the most useful thing ever, and I’d really like to get it, as I can no longer remember all the different places I’ve given my number (potential employers, school, etc.) and this would be really helpful.

However, I can’t figure out what it’s called/how much it costs/etc. Searching "call forwarding" doesn’t pull up the right thing. Does anyone know what this is, and how much it would cost from AT&T???