Once the signed offer letter exists, apply at the port of entry shortly before the start date, or have the college file Form I-129 several months ahead. Do not enter as a visitor and begin teaching while waiting.
The letter, not the semester, starts the clock
A Canadian teacher who chooses the port-of-entry route can apply days before the semester, because there is no petition to await; the risk is that a defective letter is discovered with no time to fix it, so review it in June, not August. If the college wants USCIS to adjudicate first, the petition should be filed early enough for regular processing, or with premium processing if the calendar is short. TN admission can be granted for up to three years at a time, so a one-year contract with an expected renewal should be presented as such, and a new application follows each contract extension rather than being assumed.
Summer breaks do not end the employment where the contract continues. What a teacher must not do is cross as a visitor, start preparing courses on campus, and apply later: employment before admission in TN status is unauthorized work and can affect future applications.