No. You do not need a visa (as DrJack54 already explained). You can leave and re-enter, getting a visa-exempt entry stamp.
HOWEVER... either with or without a visa, you can only get stamped in up until the expiry date of your passport in any case.
The same situation would apply even if your wife co-operated fully: immigration will not give you a one year extension ... they will only give you an extension up until the expiry date of your passport.
If you look at it properly, your only problem is that you did not apply for a new passport soon enough (the problem is *not* your wife, and the problem is *not* the number of pages remaining in your passport).