It does look like the 3M E-Z Pass adhesive strips which Amazon offers at four for $4.10 are identical to the E-Z Pass mounting strips which New York provides to E-Z Pass account holders, so they should work (if it is not convenient for you to pick up free ones near you). Assuming that you have mounting strips permanently attached to your transponder, you would just attach two more strips (Velcro side to Velcro side) to the existing strips on your transponder, peel off the backing on the new strips, and then just stick the transponder to the appropriate spot on your rental car's windshield. (The Illinois Tollway website provides instructions as to exactly where the transponder should be mounted.)

When you're done with the rental, pull your transponder off the windshield. The rental-car company probably won't care if you leave the two strips on the windshield, but I make a habit of removing them anyway.