If say, the client company offers me the job, then I apply for I-129 by mail, when new TN arrives, then there may be 1 month or 2 waiting period, right?
Can I start to work for the client company or I have to wait until new TN arrives?
That is why it is faster and simpler just to go to the border. You could start tomorrow.
Whatever flight you can get to canada is your best bet.