I was wondering, do I have to wait until pilot season ends before I call a production company to find out if there's any jobs?
Actually, the idea of "pilot season" is starting to fade more and more each and every year. Most networks and cable stations are taking more of a year round approach to their show development. Mainly so that they don't have to rely on having all of their eggs in the Fall/Autumn basket so to speak.
That said, there is certainly no reason why you should wait to contact a production office. If they don't have anything available, or the show isn't being produced yet, they'll tell you. But -- the sooner you make the connection the better it might potentially be as you never know when they'll need someone. If they have your name, resume and contact info at the ready, you might very well be the person they hire.
So, the short answer is -- don't wait. Once you get the info on a potential job opportunity -- jump on it. You never know where (or when) it might lead.