If an old calendar period loads each time you open your department's scheduler site, you are likely using a desktop shortcut or bookmark with an old web address in it. Chances are the link you have saved in that shortcut/bookmark/etc. was copied from the URL bar when you originally created the shortcut back in the day.
To correct this behavior, simply change the web address that those items (shortcut, bookmark, etc.) have saved in them to [your_site_name].pacescheduler.com/calendar
Doing so will cause your bookmark/shortcut to always load the current calendar period when clicked.
Alternatively, rather than editing an existing bookmark/shortcut, you could delete it, and easily create a new one by dragging the calendar tab into your bookmarks bar, like shown below.
Finally, if you type in the address each time, rather than using shortcuts and bookmarks, keep in mind that sometimes the browser likes to autocomplete web addresses as well, in which case you'll just want to be sure to select the standard "{your_site_name}.pacescheduler.com" address, rather than the other suggested ones.