Well, yes - as you say the problem has become apparent because there are only 12 months in the year. But I'm afraid any date comparisons in your application that contained an ambiguous date will have been wrong the whole time.
I think I'm right in saying that by default SQL Server installs with the locale set to US-English and therefore uses an MDY date format. To avoid these sorts of problems any queries used by an application should control the format of dates whether in comparisons or for display purposes. Thus dates in comparisons should always be converted to a like format, and dates for display (especially in a Web app) should be converted to a varchar version in the desired format to avoid any problems caused by the client's locale setting.
I would suggest, therefore, that you find the errant queries used by your application and correct them. If you already have data in the database(s) then changing the database and/or server settings will cause havoc you wouldn't want to contemplate...