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60-day gade period means if you lost your job (e.g., layoff), you can stay in US legally for 60 days and you can find a new job in US and transfer you TN status to the new employer within US durng this grace period, i.e., the new employer file a new I-129 petition for you to USCIS.

So, it is not your case because you want to go to the US-Canada border to get the new TN status, not do TN transfer within US. It is equal to that you go to Canada and then, reenter US using a new TN visa/status. So, you only can do it no more than 10 days prior to the new empoyment start date as I mentioned in my above response.