How to change my voicemail name on iPhone 5?

When setting up my voicemail on my iPhone 5, I was asked to record my name. I thought it was asking for a greeting for when IIm not there, but it turns out it literally just meant my name. Now it sounds stupidly cringeworthy because now people hear "voicemail from hi, it's [Name]. IIm obviously not here right now so if you still need to speak to me just call back later or send me a message. Thanks!" How do I change this? I just need it to be my name alone, not all of that garble

Your voicemail is through your cell phone carrier so call your voicemail, punch in your voicemail passcode number and listen to the menu options. There should be one for changing your greeting.