You can purchase a single re-entry permit at the immigration office or at the airport upon departure. The fee is 1,000 baht and you’ll get stamped back in within the validity of your stamp. (90 days).
You'll need to obtain a re-entry permit on your initial Non-O visa and then apply for an extension of stay on your second entry. You can submit your application for extension from 30 days prior to the due date, although some immigration offices may accept it up to 45 days in advance, and it's possible until the last day.
If you manage to extend your insurance before you travel, you will likely be granted one year from the date of entry, provided that you can document both insurance policies.
Sorry, but you’ll need to do a border bounce after renewing your health insurance in order to obtain another year of stay. After that, you may apply for yearly extensions at the local immigration office, but you will still be required to maintain approved health insurance.
Normally no issues to apply for a Non O in country on a TM.86 for a tourist visa and TM.87 for visa exemption, but a very few immigrations don’t offer this service.