I would start by drawing circles and ovals then because it has symmetry i would build it out of those circles by offsetting the other edge and finally building it out with tweaks. Manually tracing the image is in this case the sanest solution. The logo is quite simple to decostruct.
Image 1: Build out of simple shapes and offset
Once i have the symmetry and offset aligned I build and merge the shapes and offset for the other edge. And then tweak
Image 2: and what I end up with after 10 minutes of work
Edit: since i get both sides of the curve with the offset. i can just turn it into a single stroke by averaging the vertices and control points between the 2 curves (using blend).
Image 3: you can get the average with the blend tool
You can then use a stroke to guesstimate the shape.
Image 4: Using central stroke