I have a loud environment from time to time and it would be nice to be able to plug in different headphones into the phone. Are there any other plugs that will work with the phone? So far the only ones that work are the headphones that came with the phone. I have tried at least five other types of chords/plugs in the right size. Some with one or two bands, and some with up to four bands on the plug that are a little more narrow and able to get down into the plug opening.
Still, the only ones that work are the original headphones.

Is there something I'm missing, like if it's a 3.8mm jack or something crazy like that?

The original headphones will fit and there are some headphones that can work. Jack contact should be as long as possible and needs 4 areas for supporting all headset funktions. Maybe cutting a short peace of plastic away from the plug can also help. There are also adapters that are fitting.