What is the maximum extension of stay I can get after resigning from a BOI company in Thailand?

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That is what you have to ask your current employer and BOI (OSOS).

Your employer will give you documents (or support you to apply) for cancellation of your work permit and extension of stay. Those documents have the termination date of your employment. You can apply for cancellation a few days or weeks in advance. OSOS MIGHT give you a grace period, but it could be any of 0, 7, 14, or 21 days depending on the BOI company (employer) and maybe some other factors. <- This is also what you need to ask your employer AND BOI.

You need to talk with your current and new employer about the procedures for you to follow.