"the 60 days visa-exempt can evidently be extended at local immigration for 30 days"
Not so, I believe.
This is what the Thai Embassy in London is saying:-
"Tourist Visa Exemption Scheme
UK ordinary passport holders can travel to Thailand without visa for no longer than 60 days (maximum 2 times/year).
All UK travel documents (Refugee/Emergency) must apply for the visa before traveling to Thailand.
- Foreigners entering Thailand by any means under the Visa Exemption scheme are required at the port of entry to have proof of planned travel (confirmed air, train, bus, or boat tickets) to leave Thailand within 60 days of the arrival date. Otherwise, a visa must be obtained before entering Thailand."