There was a perfectly good name Undecember already: it actually occurred in 46 BCE (just after November, before Duodecember, and two months before December). The terminal difference between decem "ten" and undecim "eleven" is presumably because of the difference in distances away from the stress: so in Undecember, with stress on -cem-, this change of /e/ to /i/ would not apply.
And I don't think the existence of a thirteenth month implies that there can be more than thirteen: how many returns of the moon can you fit in one year? Okay, 46 BCE was an exception and had fifteen months.
The thirteenth month of the calendar. Bet you didn't know there were thirteen months, did you? In fact, you can technically have as many months as you want. Our Gregorian calendar only uses twelve of them. But certain calendars, such as the Hebrew or Arabian, have the potentiality for thirteen months: and to support these in computer programs, specifically the Java language, the term Undecimber was introduced.
The name derives from the Latin undecim, or eleven.