Do you know anything about The Coding Network?
Yes, I've actually used them for some interventional coding we needed to have done until I could find a permanent coder. They do charge by the chart as do most remote companies. They are reputable, and I have met them at AAPC conferences. We had no issues with their service. However, from an employee perspective, I understand that the work is not steady to start out with. You may want to check on productivity standards. If they pay you at a flat chart rate, then you can code only one chart in an hour because you're busy with the newborn, and it won't be a problem for them. However if they expect a certain number of charts per hour and you can't maintain that because of your situation, you'll need to know that. Also, I can't recall if their coders were employees or contractors. As a contractor you would be responsible for paying your own taxes, among other things. I'm not trying to discourage you, but these are things you need to think about.
I do recognize the challenge for working parents; been there, done that. Just make sure you go into this with eyes wide open, but don't devalue yourself. A certified Radiology Coder with 17 years experience should be bringing in more than a few dollars an hour. Are you on LinkedIn? Have you checked the job board on this AAPC site? Contact Rad practices in your area, and see if the (or their billing company) needs coders. Most places allow for remote workers in today's business environment, so that shouldn't be a deterrent. That way you're an employee and not a contractor. Good luck.