Retain circular shape after adding subdivisions with sub surf mod applied Anytime you need to add additional subdivisions to only certain edges of a circle (not all edges) you wind up with a shape that's no longer circular when the sub surf mod calculates the curve between the newly added vertex and the original vertices. 

Logically it seems that there should be a way to determine where that new vertex needs to be to make the resulting sub surf line appear circular. Unfortunately,  as a human, it's hard for me to pinpoint that exact location by eye.

Is there a tool or command that re-circularizes the shape once I add my subdivision to my circles edge? Am I thinking about this the wrong way such that there's an alternative path that fixes the problem without this tool?

Thanks in adv.
If I understand well, your issue is to recalculate edges/vertices to be circle shape after adding more cuts, maybe this answer could help.
Anyway if that the case, you should first outline well your circle mesh using as less as possible edges, or add circle mesh instead outlining it by hand.