We learn from his phone message that as well as teaching at Chilton, Max teaches a night class twice a week in Stamford, a city in Connecticut about an hour and a half drive south from Hartford, on the coast.
The fact that Max sees the sign to Stars Hollow on the way suggests again that it around the Wallingford area, as you would drive past Wallingford going from Hartford to Stamford using Interstate 91. It isn’t actually a turnpike though as it is toll-free at that point: the Connecticut Turnpike (I-95) goes further on from Stamford toward New Haven.
Max possibly teaches in the English department at the Stamford campus of the University of Connecticut, a public university. We learn a bit later that one of the nights he teaches class is on Fridays, so with Lorelai attending classes twice a week and Max teaching classes twice a week, it’s going to be hard for them to find a free day in the week to date.