How would I make a mob attach or follow the player seamlessly?

I think the title says it, but basically, in my mod I'm adding hair. I want an entity that seamlessly attaches itself to the player and follows them around. Kind of like a parrot or something. I don't know if this can be done with armor, I think I tried but the models didn't work with it, so I'm thinking an entity would be easier. Could someone explain how to do this with procedures? I don't want it to look like it's constantly teleporting to me like with a command on repeat, so any help would be greatly appreciated!