On certain special Shabboses, like the four parshiyot, there is a special torah reading that replaces Maftir after the weekly parsha is read. Is there a name for this extra not-from-the-weekly-parsha Maftir reading?
I have never heard of a name that refers specifically to the category of "maftir readings that are not from the weekly sedra," and I do not believe that such a name exists. In my experience, such a reading is usually just referred to as a "special maftir reading" (as you mentioned in your comment).
I will try to clarify some of the confusion that has appeared in the comments on this post and answers about the word "maftir". The maftir is the person who reads the haftarah. Both words come from the same root (פ.ט.ר.). Maftir is the present-tense third-person form of the word in binyan Hif'il. I don't want to get too deep into a grammar lesson, but Binyan Hif'il is used for causative verbs (i.e. performing an action). The third-person present-tense hif'il verb form is also often used to refer to a person who performs that action, so that's why the person who reads the haftarah is called the maftir.
The maftir is given the aliya before the haftarah. That aliya is therefore called the maftir aliya (which is sometimes abbreviated as simply maftir). Most Shabbatot, the maftir aliya is simply a repeat of the last few verses from that week's sedra; however, on certain special Shabbatot (e.g. the Four Parshiot) as well as holidays there is a "special maftir" from a completely different part of the Torah. That is the reading that your quesiton asks about and for which I do not believe there is any special name.
I think you may be confused regarding the term "maftir" and how it works.
"Maftir" refers to the "additional" aliyah that occurs after the other 7 required aliyot on Shabbat. There is always a maftir, so there is no additional maftir. It is merely that one maftir is substituted in place of the standard weekly maftir.
Some people call the Haftarah "maftir". The word "maftir" refers to the last reading from the Torah, whereas "haftarah" refers to the reading from the prophets. The person who reads the haftarah receives the maftir aliyah.
On mincha of fast days and Tish'a B'av morning, the "maftir" aliyah is not an extra one, but rather is the 3rd aliyah. That's the only exception to not adding an extra one as we have on Shabbat and Yom Tov.
Sometimes you hear someone say "I have maftir Yonah" if he is reading that section on Yom Kippur afternoon. That's become parlance, I suppose, but as I explained above, he should say, "I have haftarah Yonah".