Can you have twi jobs under a TN visa?

Thank you all for quick replies!!

When I apply for another job, I will ask for a job offer letter to the employer . Do I have to cross the border in order to obtain the name of the employer under the current TN visa? Or do I receive another TN visa paper?

I still have more questions.
What would be responsibilities for the employer to support the TN visa? In short, are there any financial or other responsibilities which may stop supporting TN visa?

I am thinking of applying to a small clinic as a second job. This is a really small clinic and they have never supported TN/Green card before. I am just wondering if there are any difficulties to obtain TN visa from them.

Thank you for all your support!