I don't believe that they perform better when they have a new contract. I usually wait for the next season to sign them again. No matter how good a player is, I never know if I find a better one during the season and I decide to sell the old one.
And something else. If you decide to sign them the first day of the next season , you may want to stop adding skill points to them in the end of this season in order to keep their price as low as possible.