I don't think Kirk immediately knew that Finney was missing. If the ion pod was the sole reason to enter the storm, then after it was jettisoned Kirk would have concentrated on getting the ship out of the ion storm. If the Enterprise's regular sensor could gather additional readings, Kirk would have stayed in the storm longer, before leaving. When would Finney have been missed? He should have called in as soon as he was out, perhaps to Uhura, then reported to his red alert station. Kirk would have started by simply calling him on the shipwide intercom A shipwide search would have had to wait until the ship was both out of the storm and secure.
I like the idea that as records officer Finney was the ship's I.T. geek.
We've all been assuming (me too) that the ion pod was small, but what if it wasn't? What if the pod was very large? Something that was attached to the length of the secondary hull, a dozen meters across and fifty or sixty meters long. When Kirk ordered warp one, then told engineering to increase power by one third, that may have been to compensate for the presence of the pod. The size of the pod interfered with the Enterprise's ability to maneuver properly, created torque, slowed her down. Kirk jettisoned the pod not because it was becoming contaminated, but because it mere presence endangered the ship.
When jettisoned the combination of the Enterprise being at warp, leaving the warp envelope, and the surounding ion storm would have torn the pod apart.