@Alejandro **********
So it's an extension of stay. Here is the process you will have to perform:
1. You will have to cancel your existing "extension of stay" stamp. You will take a letter from the school saying you are no longer a student. This will have to be "dated" in advance for 14 days (for Bangkok), OR 21 days (for Chiang Mai). Other immigration offices have different requirements.
2. You, and your employer, will have to apply for an in-country Non-B visa based on employment. The employer would know what type of paperwork is required.
3. You will be issued an in-country Non-B Visa which is valid for 90 days.
4. When you have 30 days or less remaining (45 days at some immigration offices), you can apply for a one year extension of stay based on employment.
HOWEVER, this depends on the immigration office you are using, and the employer, and not all immigration offices sell in-country Non-B visas, and not all employers can get an in-country Non-B visa.