, right, once she leaves and re-enters Thailand, then she is no longer "currently" overstaying. She can apply for an in-country initial visa with the correct documents as long as she has 15 "working" days (usually 21 calendar days) at the Chiang Mai office.
, as long as your daughter's nationality is eligible for visa-exempt, one day trip to re-enter Thailand on visa-exempt is not a problem. Then she will be able to apply for an in-country visa.
But It's you and the employer's decision where you want to apply for her initial Non-O visa, because required paperwork at consulate and immigration office are different.
, OP is not talking about how to stay. OP's daughter has to clear her overstaying by leaving and re-entering Thailand to be able to apply for a proper visa.
Usually medical extension must be applied before you go into overstaying. Children under 15yo is not fined for overstaying but get overstay stamp.
You cannot bounce at any northern Myanmar borders.
Depending on the border checkpoint, some borders just don't let you leave if you are overstaying for more than specific days. In this case, medical cert cannot do anything.
You might need to stay for 1-2 nights in Laos if you go Laos borders with Chiang Rai.
You better talk with services if you want to bounce at the land borders. (no recommendations from this group though)