Team schedules are defaulted to automatically start on July 1st each year. If you want to modify that, you can go to the Team Classification section from the Athletics Manager Dashboard and adjust the Event Year dropdown menu. Set the dropdown to the month range that is appropriate for your needs.
Note: Upcoming events that fall into the next Event Year by virtue of the settings in this dropdown menu will not be displayed unless the "Publish events in future Event Years" checkbox is selected for the individual teams (in the Team Properties section of the team). For example, if you choose to stay with a July through June year and view events scheduled for June of the current year, you will not see any events scheduled beyond the end of June. Once the new event year starts, then the full year of upcoming events will be visible.