Trail Renderer which ignores parent movement

I'm trying to create an effect where a whisp orbits the player and leaves a circular trail around them. I created a child object with a trail renderer attached and a script which orbits the child object around the local coordinates of the player.

The trail effect is incorrect however. When the player moves, this also creates a trail, leading to a wobbly drawn circle. What I actually want to happen is for the orbiting to create a perfect circle trail, and for player movement to simply shift the whole trail over by that amount as if it is in a container. Is this possible?

In searching for a solution, I came across the following question which has an answer that explains how to achieve this effect for a particle system. It mentions the trail renderer as well but does not elaborate on a solution for this component.