Not sure where my company got this logo, but they've had it for years and years.

OGS Logo

There is no vector image of it. I'm a photoshop guy and I know a little Ai. I tried doing the image trace option and it didn't work too well, very jaggedy. I know it's not the greatest of images, but I have no other options.

Would this be something that you Ai guys would just whip out the pen tool and trace around it, I'm not very proficient at it, and would have to spend a decent amount of time learning it. Or perhaps I didn't use the image trace tool properly?