Your 2.6″ tires measure 2.4″ because of your rim width. If you want 2.6″ tires on the bike (not just in name), you will need a wider rim. Maybe 5mm wider.
I’d hazard a guess and say the 2.35″ tires will not be much narrower than the 2.6″ tires when mounted on your rims. The main difference will be the 2.6″ tires mounted will be taller (larger diameter) than the 2.35″ tires.