A non immigrant B visa is not the same as a tourist visa...you don't get extensions for tourism purposes.

A 3 month non-immigrant B visa is delivered to foreigners working in Thailand... usually you require a letter from your employer and a signed contract to get one. Only when you have a three month visa can you apply for a work permit. Once you have a work permit you then get a one year non Immigrant B visa.

While you are on a 3 month non imm B visa you are free to stay in the country until it expires even if you have left your job. This isn't the case with the year long visa. It's totally dependent on your work contract. If you quit your job or get sacked then you have 7 days to leave the country.

In your case you will not be eligible for an extension of your non-Immigrant B visa because you will no longer be working for your school. You will have to leave the country when it expires anyway.

Depending on how long you want to stay after that you can either get a free 30 day entry stamp or a 60 day tourist visa from a Thai consulate outside Thailand.

__________________
“Not all those who wander are lost.” Tolkien.
Life is not measured by the number of breaths we take, but by the moments that take our breath away.