Not trying to scare you, just noting that I have had at least one friend on TN's who tried to bridge their status without filing a proper change in status ( ie. to B-2 as soon as you quit/are let go from your old employer, to be eligible to stay without packing up your apartment) and were banned from re-entry for 2 years.
I have had several TN's over the past 16 years and the only trouble I have had was primary border guard overreach once or twice (they felt I had been in the USA long enough, arbitrarily. Secondary review fixed that.)
I did not realize I could file a change in status if my old job concluded before I got a new one. In 2003, I finished up with a company, and my last, "Paid Day" as an employee was September 26th, but I wasn't required to come into the office for the last two weeks - so my last day in the office was around September 12th, a friday. By Sunday, September 14th, I had emptied my apartment, packed up all my possessions, and had returned to Canada.
The only evidence I have submitted to a half dozen employers of my right to work in the United States (because it's all I've ever been given) - is a cardboard I-94, filled out by me, no less, with a faint, blue stamp and date, and a (once again handwritten, this time by ICE agent) "Class: TN, Until: 07 Oct 2015". No Serial Number, no tracking number - just a stamp and visa class.