Tornado season does not have a set start and end date. If the conditions are right for a tornado, they can occur anytime throughout the year. In fact, it depends on where you live in the US.
If you live more to the southern portions of Tornado Alley, tornadoes are more likely to occur in the late winter and early spring. The farther you live north, the more likely tornadoes are to occur in the latter parts of summer.